{"id":1363,"date":"2022-01-28T19:34:55","date_gmt":"2022-01-28T19:34:55","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/trigonometry\/chapter\/geometric-sequences\/"},"modified":"2022-01-28T19:34:55","modified_gmt":"2022-01-28T19:34:55","slug":"geometric-sequences","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/trigonometry\/chapter\/geometric-sequences\/","title":{"raw":"Geometric Sequences","rendered":"Geometric Sequences"},"content":{"raw":"\n\n\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Find the common ratio for a geometric sequence.<\/li>\n \t<li>List the terms of a geometric sequence.<\/li>\n \t<li>Use a recursive formula for a geometric sequence.<\/li>\n \t<li>Use an explicit formula for a geometric sequence.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137639564\">Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section, we will review sequences that grow in this way.<\/p>\n\n<div id=\"fs-id1165137715137\" class=\"bc-section section\">\n<h3>Finding Common Ratios<\/h3>\n<p id=\"fs-id1165137894517\">The yearly salary values described form a <strong>geometric sequence<\/strong> because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the <strong>common ratio<\/strong>. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.<\/p>\n<span id=\"fs-id1165137730802\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154844\/CNX_Precalc_Figure_11_03_001.jpg\" alt=\"A sequence , {1, 6, 36, 216, 1296, ...} that shows all the numbers have a common ratio of 6.\"><\/span>\n<div id=\"fs-id1165137749987\" class=\"textbox key-takeaways\">\n<h3>Definition of a Geometric Sequence<\/h3>\n<p id=\"fs-id1165137855264\">A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If [latex]{a}_{1}[\/latex] is the initial term of a geometric sequence and [latex]r[\/latex] is the common ratio, the sequence will be<\/p>\n\n<div id=\"eip-246\" class=\"unnumbered aligncenter\">[latex]\\left\\{{a}_{1}, \\,{a}_{1}r,\\,{a}_{1}{r}^{2},\\,{a}_{1}{r}^{3},\\,...\\right\\}.[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137417317\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135388542\"><strong>Given a set of numbers, determine if they represent a geometric sequence.<\/strong><\/p>\n\n<ol id=\"fs-id1165137399185\" type=\"1\">\n \t<li>Divide each term by the previous term.<\/li>\n \t<li>Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_03_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137594043\">\n<div id=\"fs-id1165137736734\">\n<h3>Finding Common Ratios<\/h3>\n<p id=\"fs-id1165137462087\">Is the sequence geometric? If so, find the common ratio.<\/p>\n\n<ol id=\"fs-id1165137911197\" type=\"a\">\n \t<li>[latex]1\\text{,}\\,2\\text{,}\\,4\\text{,}\\,8\\text{,}\\,16\\text{,}\\,...[\/latex]<\/li>\n \t<li>[latex]48\\text{,}\\,12\\text{,}\\,4\\text{, }2\\text{,}\\,...[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137430766\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137430766\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137430766\"]Divide each term by the previous term to determine whether a common ratio exists.\n<ol id=\"fs-id1165137410224\" type=\"a\">\n \t<li>[latex]\\begin{array}{llllllllll}\\frac{2}{1}=2\\hfill &amp; \\hfill &amp; \\hfill &amp; \\frac{4}{2}=2\\hfill &amp; \\hfill &amp; \\hfill &amp; \\frac{8}{4}=2\\hfill &amp; \\hfill &amp; \\hfill &amp; \\frac{16}{8}=2\\hfill \\end{array}[\/latex]\n<p id=\"fs-id1165137837038\">The sequence is geometric because there is a common ratio. The common ratio is 2.<\/p>\n<\/li>\n \t<li>[latex]\\begin{array}{lllllll}\\frac{12}{48}=\\frac{1}{4}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\frac{4}{12}=\\frac{1}{3}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\frac{2}{4}=\\frac{1}{2}\\hfill \\end{array}[\/latex]\n<p id=\"fs-id1165137603152\">The sequence is not geometric because there is not a common ratio.<\/p>\n<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id1165137667273\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137448166\">The graph of each sequence is shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_03_002\">(Figure)<\/a>. It seems from the graphs that both (a) and (b) appear have the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this interpretation holds, but (b) is not.<\/p>\n\n<div id=\"CNX_Precalc_Figure_11_03_002\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154847\/CNX_Precalc_Figure_11_03_002.jpg\" alt=\"Graph of two sequences where graph (a) is geometric and graph (b) is exponential.\" width=\"975\" height=\"286\"> <strong>Figure 1.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137742586\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137611774\"><strong>If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the common ratio?<\/strong><\/p>\n<p id=\"fs-id1165137803401\"><em>No. If you know that the sequence is geometric, you can choose any one term in the sequence and divide it by the previous term to find the common ratio.<\/em><\/p>\n\n<\/div>\n<div id=\"fs-id1165135241213\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_01\">\n<div id=\"fs-id1165137506295\">\n<p id=\"fs-id1165135408468\">Is the sequence geometric? If so, find the common ratio.<\/p>\n\n<div id=\"eip-id1491662\" class=\"unnumbered\">[latex]5,10,15,20,...[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137783622\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137783622\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137783622\"]\n<p id=\"fs-id1165137667314\">The sequence is not geometric because [latex]\\frac{10}{5}\\ne \\frac{15}{10}[\/latex].<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-234\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_02\">\n<div id=\"fs-id1165137645227\">\n<p id=\"fs-id1165137462478\">Is the sequence geometric? If so, find the common ratio.<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"415842\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"415842\"]\n<div id=\"fs-id1165137645227\">\n<div id=\"eip-id2227162\" class=\"unnumbered\">[latex]100,20,4,\\frac{4}{5},...[\/latex]<\/div>\n<\/div>\n<div>\n\nThe sequence is geometric. The common ratio is [latex]\\frac{1}{5}[\/latex].\n\n<\/div>\n[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137433726\" class=\"bc-section section\">\n<h3>Writing Terms of Geometric Sequences<\/h3>\nNow that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4,[\/latex] we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.\n<div id=\"eip-245\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)=-128\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137472075\">The first four terms are[latex]\\left\\{\u20132\\text{, }\u20138\\text{, }\u201332\\text{, }\u2013128\\right\\}.[\/latex]<\/p>\n\n<div id=\"fs-id1165137939702\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137473764\"><strong>Given the first term and the common factor, find the first four terms of a geometric sequence.<\/strong><\/p>\n\n<ol id=\"fs-id1165137409884\" type=\"1\">\n \t<li>Multiply the initial term, [latex]{a}_{1},[\/latex] by the common ratio to find the next term, [latex]{a}_{2}.[\/latex]<\/li>\n \t<li>Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n \t<li>Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_03_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137573089\">\n<div id=\"fs-id1165137462497\">\n<h3>Writing the Terms of a Geometric Sequence<\/h3>\n<p id=\"fs-id1165137640196\">List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=\u20132.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137758063\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137758063\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137758063\"]\n<p id=\"fs-id1165137811084\">Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}.[\/latex] Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3},[\/latex]\nand so on.<\/p>\n\n<div id=\"eip-id1165135160893\" class=\"unnumbered\">[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137560200\">The first four terms are [latex]\\left\\{5,\u201310,20,\u201340\\right\\}.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137416531\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_03\">\n<div id=\"fs-id1165137619295\">\n\nList the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}.[\/latex]\n\n<\/div>\n<div id=\"fs-id1165137655223\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137655223\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137655223\"]\n<p id=\"fs-id1165137560204\">[latex]\\left\\{18,6,2,\\frac{2}{3},\\frac{2}{9}\\right\\}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135694573\" class=\"bc-section section\">\n<h3>Using Recursive Formulas for Geometric Sequences<\/h3>\n<p id=\"fs-id1165137641691\">A <span class=\"no-emphasis\">recursive formula<\/span> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n\n<div id=\"fs-id1165137448332\" class=\"textbox key-takeaways\">\n<h3>Recursive Formula for a Geometric Sequence<\/h3>\n<p id=\"fs-id1165137463184\">The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n\n<div id=\"fs-id1165137662634\">[latex]{a}_{n}=r{a}_{n-1},n\\ge 2[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135207482\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137559120\"><strong>Given the first several terms of a geometric sequence, write its recursive formula.<\/strong><\/p>\n\n<ol id=\"fs-id1165137442323\" type=\"1\">\n \t<li>State the initial term.<\/li>\n \t<li>Find the common ratio by dividing any term by the preceding term.<\/li>\n \t<li>Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_03_03\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165137640113\">\n<h3>Using Recursive Formulas for Geometric Sequences<\/h3>\n<p id=\"fs-id1165137470342\">Write a recursive formula for the following geometric sequence.<\/p>\n\n<div id=\"eip-id1165132059630\" class=\"unnumbered\">[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135190538\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135190538\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135190538\"]\n<p id=\"fs-id1165137645992\">The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n\n<div id=\"eip-id1165134410969\" class=\"unnumbered\">[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\n<p id=\"fs-id1165137539505\">Substitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}.[\/latex]<\/p>\n\n<div id=\"eip-id1165135694281\" class=\"unnumbered\">[latex]\\begin{array}{l}{a}_{n}=r{a}_{n-1}\\\\ {a}_{n}=1.5{a}_{n-1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137629524\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137409471\">The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_03_003\">(Figure)<\/a><\/p>\n\n<div id=\"CNX_Precalc_Figure_11_03_003\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154849\/CNX_Precalc_Figure_11_03_003.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"215\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137804164\" class=\"textbox key-takeaways\">\n<p id=\"fs-id1165137594708\"><strong>Do we have to divide the second term by the first term to find the common ratio?<\/strong><\/p>\n<p id=\"fs-id1165135593534\"><em>No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n\n<\/div>\n<div id=\"fs-id1165137461314\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_05\">\n<div id=\"fs-id1165137538955\">\n<p id=\"fs-id1165137418734\">Write a recursive formula for the following geometric sequence.<\/p>\n\n<div id=\"eip-id1264736\" class=\"unnumbered\">[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137605332\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137605332\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137605332\"]\n<p id=\"fs-id1165137779221\">[latex]\\begin{array}{l}{a}_{1}=2\\\\ {a}_{n}=\\frac{2}{3}{a}_{n-1}\\text{ for }n\\ge 2\\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137656805\" class=\"bc-section section\">\n<h3>Using Explicit Formulas for Geometric Sequences<\/h3>\n<p id=\"fs-id1165137507452\">Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n\n<div class=\"unnumbered\">[latex]{a}_{n}={a}_{1}{r}^{n-1}[\/latex]<\/div>\n<p id=\"fs-id1165137410536\">Let\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}.[\/latex] This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n\n<div id=\"eip-422\" class=\"unnumbered aligncenter\">[latex]{a}_{n}=18\u00b7{2}^{n-1}[\/latex]<\/div>\n<p id=\"fs-id1165137473099\">The graph of the sequence is shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_03_004\">(Figure)<\/a>.<\/p>\n\n<div id=\"CNX_Precalc_Figure_11_03_004\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154852\/CNX_Precalc_Figure_11_03_004.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"440\"> <strong>Figure 3.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165137647200\" class=\"textbox key-takeaways\">\n<h3>Explicit Formula for a Geometric Sequence<\/h3>\n<p id=\"fs-id1165137835426\">The <em>n<\/em>th term of a geometric sequence is given by the <span class=\"no-emphasis\">explicit formula<\/span>:<\/p>\n\n<div id=\"fs-id1165137461602\">[latex]{a}_{n}={a}_{1}{r}^{n-1}[\/latex]<\/div>\n<\/div>\n<div id=\"Example_11_03_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137453521\">\n<div id=\"fs-id1165137811825\">\n<h3>Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n<p id=\"fs-id1165137874642\">Given a geometric sequence with[latex]\\,{a}_{1}=3\\,[\/latex]and[latex]\\,{a}_{4}=24,\\,[\/latex]find [latex]{a}_{2}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137408599\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137408599\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137408599\"]The sequence can be written in terms of the initial term and the common ratio[latex]\\,r.[\/latex]\n<div id=\"eip-id1165134060418\" class=\"unnumbered\">[latex]3,3r,3{r}^{2},3{r}^{3},...[\/latex]<\/div>\n<p id=\"fs-id1165137432346\">Find the common ratio using the given fourth term.<\/p>\n\n<div id=\"eip-id1165137444653\" class=\"unnumbered\">[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n-1}\\hfill &amp; \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill &amp; \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\,\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill &amp; \\text{Substitute }24\\text{ for}\\,{a}_{4}\\hfill \\\\ \\,\\,\\,8={r}^{3}\\hfill &amp; \\text{Divide}\\hfill \\\\ \\,\\,\\,r=2\\hfill &amp; \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137455556\">Find the second term by multiplying the first term by the common ratio.<\/p>\n\n<div id=\"eip-id1165135426318\" class=\"unnumbered\">[latex]\\begin{array}{ll}{a}_{2}\\hfill &amp; =2{a}_{1}\\hfill \\\\ \\hfill &amp; =2\\left(3\\right)\\hfill \\\\ \\hfill &amp; =6\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137725528\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137789060\">The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137444993\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_04\">\n<div id=\"fs-id1165137459454\">\n<p id=\"fs-id1165137653766\">Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex], find [latex]{a}_{6}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137447458\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137447458\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137447458\"]\n<p id=\"fs-id1165137532391\">[latex]{a}_{6}=16,384[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_11_03_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137611787\">\n<div id=\"fs-id1165135255552\">\n<h3>Writing an Explicit Formula for the <em>n<\/em>th Term of a Geometric Sequence<\/h3>\n<p id=\"fs-id1165137414859\">Write an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n\n<div id=\"eip-id1165134113698\" class=\"unnumbered\">[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137911060\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137911060\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137911060\"]The first term is 2. The common ratio can be found by dividing the second term by the first term.\n<div id=\"eip-id1165134085791\" class=\"unnumbered\">[latex]\\frac{10}{2}=5[\/latex]<\/div>\n<p id=\"fs-id1165137407052\">The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n\n<div id=\"eip-id1165137558360\" class=\"unnumbered\">[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n-1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n-1}\\hfill \\end{array}[\/latex]<\/div>\nThe graph of this sequence in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_03_005\">(Figure)<\/a> shows an exponential pattern.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154854\/CNX_Precalc_Figure_11_03_005.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"290\"> <strong>Figure 4.<\/strong>[\/caption]\n<p id=\"fs-id1165137841605\">[\/hidden-answer]<span id=\"fs-id1165137460332\"><\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137600603\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_06\">\n<div id=\"fs-id1165137762502\">\n<p id=\"fs-id1165137528367\">Write an explicit formula for the following geometric sequence.<\/p>\n\n<div id=\"eip-id1673413\" class=\"unnumbered\">[latex]\\left\\{\u20131\\text{, }3\\text{, }\u20139\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137532482\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137532482\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137532482\"]\n<p id=\"fs-id1165137435217\">[latex]{a}_{n}=-{\\left(-3\\right)}^{n-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135251448\" class=\"bc-section section\">\n<h3>Solving Application Problems with Geometric Sequences<\/h3>\n<p id=\"fs-id1165137436841\">In real-world scenarios involving arithmetic sequences, we may need to use an initial term of [latex]{a}_{0}[\/latex] instead of [latex]{a}_{1}.\\,[\/latex]In these problems, we can alter the explicit formula slightly by using the following formula:<\/p>\n\n<div id=\"fs-id1165137455099\" class=\"unnumbered aligncenter\">[latex]{a}_{n}={a}_{0}{r}^{n}[\/latex]<\/div>\n<div id=\"Example_11_03_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137675504\">\n<div id=\"fs-id1165135207496\">\n<h3>Solving Application Problems with Geometric Sequences<\/h3>\n<p id=\"fs-id1165137724701\">In 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.<\/p>\n\n<ol id=\"fs-id1165137724573\" type=\"a\">\n \t<li>Write a formula for the student population.<\/li>\n \t<li>Estimate the student population in 2020.<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137603324\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137603324\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137603324\"]\n<ol id=\"fs-id1165137812960\" type=\"a\">\n \t<li>\n<p id=\"fs-id1165137475561\">The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04.<\/p>\n<p id=\"fs-id1165135345013\">Let [latex]P[\/latex] be the student population and [latex]n[\/latex] be the number of years after 2013. Using the explicit formula for a geometric sequence we get<\/p>\n\n<div id=\"eip-id1165134384459\" class=\"unnumbered\">[latex]{P}_{n} =284\\cdot {1.04}^{n}[\/latex]<\/div><\/li>\n \t<li>\n<p id=\"fs-id1165137472563\">We can find the number of years since 2013 by subtracting.<\/p>\n\n<div id=\"eip-id1165137664601\" class=\"unnumbered\">[latex]2020-2013=7[\/latex]<\/div>\n<p id=\"fs-id1165137453691\">We are looking for the population after 7 years. We can substitute 7 for [latex]n[\/latex] to estimate the population in 2020.<\/p>\n\n<div id=\"eip-id1165135699078\" class=\"unnumbered\">[latex]{P}_{7}=284\\cdot {1.04}^{7}\\approx 374[\/latex]<\/div>\n<p id=\"fs-id1165137528682\">The student population will be about 374 in 2020.[\/hidden-answer]<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137768542\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_07\">\n<div id=\"fs-id1165137410731\">\n<p id=\"fs-id1165137834028\">A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.<\/p>\n\n<ol id=\"eip-id1165134294794\" type=\"a\">\n \t<li>Write a formula for the number of hits.<\/li>\n \t<li>Estimate the number of hits in 5 weeks.<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137455278\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137455278\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137455278\"]\n<ol id=\"eip-id1165135538601\" type=\"a\">\n \t<li>[latex]{P}_{n} = 293\\cdot 1.026{a}^{n}[\/latex]<\/li>\n \t<li>The number of hits will be about 333.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137443318\" class=\"precalculus media\">\n<p id=\"fs-id1165137817563\">Access these online resources for additional instruction and practice with geometric sequences.<\/p>\n\n<ul>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/geometricseq\">Geometric Sequences<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/sequencetype\">Determine the Type of Sequence<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/sequenceformula\">Find the Formula for a Sequence<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135264797\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165133155748\" summary=\"..\">\n<tbody>\n<tr>\n<td>recursive formula for [latex]nth[\/latex] term of a geometric sequence<\/td>\n<td>[latex]{a}_{n}=r{a}_{n-1},n\\ge 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>explicit formula for[latex]\\,nth\\,[\/latex]term of a geometric sequence<\/td>\n<td>[latex]{a}_{n}={a}_{1}{r}^{n-1}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137611739\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135189853\">\n \t<li>A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.<\/li>\n \t<li>The constant ratio between two consecutive terms is called the common ratio.<\/li>\n \t<li>The common ratio can be found by dividing any term in the sequence by the previous term. See <a class=\"autogenerated-content\" href=\"#Example_11_03_01\">(Figure)<\/a>.<\/li>\n \t<li>The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See <a class=\"autogenerated-content\" href=\"#Example_11_03_02\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_11_03_04\">(Figure)<\/a>.<\/li>\n \t<li>A recursive formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]\\,{a}_{n}=r{a}_{n\u20131}\\,[\/latex]for [latex]n\\ge 2[\/latex].<\/li>\n \t<li>As with any recursive formula, the initial term of the sequence must be given. See <a class=\"autogenerated-content\" href=\"#Example_11_03_03\">(Figure)<\/a>.<\/li>\n \t<li>An explicit formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]\\,{a}_{n}={a}_{1}{r}^{n\u20131}.[\/latex] See <a class=\"autogenerated-content\" href=\"#Example_11_03_05\">(Figure)<\/a>.<\/li>\n \t<li>In application problems, we sometimes alter the explicit formula slightly to [latex]\\,{a}_{n}={a}_{0}{r}^{n}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_11_03_06\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137695322\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137603819\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137415276\">\n<div id=\"fs-id1165137601852\">\n<p id=\"fs-id1165137460182\">What is a geometric sequence?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137413921\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137413921\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137413921\"]\n<p id=\"fs-id1165137419043\">A sequence in which the ratio between any two consecutive terms is constant.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135386446\">\n<div id=\"fs-id1165137430484\">\n<p id=\"fs-id1165137430485\">How is the common ratio of a geometric sequence found?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137651855\">\n<div id=\"fs-id1165137871059\">\n<p id=\"fs-id1165137666549\">What is the procedure for determining whether a sequence is geometric?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137571584\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137571584\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137571584\"]\n<p id=\"fs-id1165137645826\">Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137786269\">\n<div id=\"fs-id1165137698332\">\n<p id=\"fs-id1165137767192\">What is the difference between an arithmetic sequence and a geometric sequence?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137806166\">\n<div id=\"fs-id1165137812741\">\n\nDescribe how exponential functions and geometric sequences are similar. How are they different?\n\n<\/div>\n<div id=\"fs-id1165137410439\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137410439\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137410439\"]\n<p id=\"fs-id1165137735983\">Both geometric sequences and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an exponential function must be positive.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135185333\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\nFor the following exercises, find the common ratio for the geometric sequence.\n<div id=\"fs-id1165137656402\">\n<div id=\"fs-id1165137628207\">\n<p id=\"fs-id1165137643000\">[latex]1,3,9,27,81,...[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137698185\">\n<div id=\"fs-id1165137433830\">\n<p id=\"fs-id1165137936860\">[latex]-0.125,0.25,-0.5,1,-2,...[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137560236\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137560236\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137560236\"]\n<p id=\"fs-id1165137626648\">The common ratio is [latex]-2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137696255\">\n<div id=\"fs-id1165137666722\">\n<p id=\"fs-id1165137925247\">[latex]-2,-\\frac{1}{2},-\\frac{1}{8},-\\frac{1}{32},-\\frac{1}{128},...[\/latex]<\/p>\n\n<\/div>\n<\/div>\nFor the following exercises, determine whether the sequence is geometric. If so, find the common ratio.\n<div id=\"fs-id1165137533956\">\n<div id=\"fs-id1165137643508\">\n<p id=\"fs-id1165137725448\">[latex]-6,-12,-24,-48,-96,...[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137660469\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137660469\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137660469\"]\n<p id=\"fs-id1165137550725\">The sequence is geometric. The common ratio is 2.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137462907\">\n<div id=\"fs-id1165137452609\">\n<p id=\"fs-id1165137803618\">[latex]5,5.2,5.4,5.6,5.8,...[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137823090\">\n<div id=\"fs-id1165137433632\">\n<p id=\"fs-id1165137823320\">[latex]-1,\\frac{1}{2},-\\frac{1}{4},\\frac{1}{8},-\\frac{1}{16},...[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134047692\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134047692\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134047692\"]\n<p id=\"fs-id1165137911673\">The sequence is geometric. The common ratio is [latex]-\\frac{1}{2}.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137935634\">\n<div id=\"fs-id1165137517318\">\n<p id=\"fs-id1165137723782\">[latex]6,8,11,15,20,...[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137841659\">\n<div id=\"fs-id1165137841662\">\n<p id=\"fs-id1165137542166\">[latex]0.8,4,20,100,500,...[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137849116\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137849116\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137849116\"]\n<p id=\"fs-id1165137849119\">The sequence is geometric. The common ratio is [latex]5.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137578676\">For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.<\/p>\n\n<div id=\"fs-id1165135175313\">\n<div id=\"fs-id1165137566099\">\n<p id=\"fs-id1165137566101\">[latex]\\begin{array}{cc}{a}_{1}=8,&amp; r=0.3\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447376\">\n<div id=\"fs-id1165137539733\">\n<p id=\"fs-id1165135172312\">[latex]\\begin{array}{cc}{a}_{1}=5,&amp; r=\\frac{1}{5}\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137444326\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137444326\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137444326\"]\n<p id=\"fs-id1165135149206\">[latex]5,1,\\frac{1}{5},\\frac{1}{25},\\frac{1}{125}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135152167\">For the following exercises, write the first five terms of the geometric sequence, given any two terms.<\/p>\n\n<div id=\"fs-id1165135195397\">\n<div id=\"fs-id1165137898879\">\n<p id=\"fs-id1165135666802\">[latex]\\begin{array}{cc}{a}_{7}=64,&amp; {a}_{10}\\end{array}=512[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137694069\">\n<p id=\"fs-id1165137806803\">[latex]\\begin{array}{cc}{a}_{6}=25,&amp; {a}_{8}\\end{array}=6.25[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137405237\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137405237\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137405237\"]\n<p id=\"fs-id1165137405239\">[latex]800,400,200,100,50[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137889906\">For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.<\/p>\n\n<div id=\"fs-id1165137635538\">\n<div id=\"fs-id1165137723274\">\n<p id=\"fs-id1165137529044\">The first term is [latex]2,[\/latex] and the common ratio is [latex]3.[\/latex] Find the 5<sup>th<\/sup> term.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193320\">\n<div id=\"fs-id1165137557071\">\n<p id=\"fs-id1165137602820\">The first term is 16 and the common ratio is [latex]-\\frac{1}{3}.[\/latex] Find the 4<sup>th<\/sup> term.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135173567\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135173567\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135173567\"]\n<p id=\"fs-id1165135173569\">[latex]{a}_{4}=-\\frac{16}{27}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137460391\">For the following exercises, find the specified term for the geometric sequence, given the first four terms.<\/p>\n\n<div id=\"fs-id1165137415713\">\n<div id=\"fs-id1165137772362\">\n<p id=\"fs-id1165135174888\">[latex]{a}_{n}=\\left\\{-1,2,-4,8,...\\right\\}.[\/latex] Find [latex]{a}_{12}.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137727503\">\n<div id=\"fs-id1165135191604\">\n<p id=\"fs-id1165135187574\">[latex]{a}_{n}=\\left\\{-2,\\frac{2}{3},-\\frac{2}{9},\\frac{2}{27},...\\right\\}.[\/latex] Find [latex]{a}_{7}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137450967\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137450967\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137450967\"]\n<p id=\"fs-id1165137450969\">[latex]{a}_{7}=-\\frac{2}{729}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137642161\">For the following exercises, write the first five terms of the geometric sequence.<\/p>\n\n<div id=\"fs-id1165137849215\">\n<div id=\"fs-id1165137811176\">\n<p id=\"fs-id1165137476786\">[latex]\\begin{array}{cc}{a}_{1}=-486,&amp; {a}_{n}=-\\frac{1}{3}\\end{array}{a}_{n-1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135195370\">\n<div id=\"fs-id1165135195372\">\n<p id=\"fs-id1165134108412\">[latex]\\begin{array}{cc}{a}_{1}=7,&amp; {a}_{n}=0.2{a}_{n-1}\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137922652\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137922652\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137922652\"]\n<p id=\"fs-id1165135381368\">[latex]7,1.4,0.28,0.056,0.0112[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137698447\">For the following exercises, write a recursive formula for each geometric sequence.<\/p>\n\n<div id=\"fs-id1165137771764\">\n<div id=\"fs-id1165135496654\">\n<p id=\"fs-id1165135496656\">[latex]{a}_{n}=\\left\\{-1,5,-25,125,...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137679367\">\n<div id=\"fs-id1165137679369\">\n<p id=\"fs-id1165137599938\">[latex]{a}_{n}=\\left\\{-32,-16,-8,-4,...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137422756\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137422756\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137422756\"]\n<p id=\"fs-id1165137653539\">[latex]\\begin{array}{cc}a{}_{1}=-32,&amp; {a}_{n}=\\frac{1}{2}{a}_{n-1}\\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447992\">\n<div id=\"fs-id1165135543366\">\n<p id=\"fs-id1165135543368\">[latex]{a}_{n}=\\left\\{14,56,224,896,...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137895431\">\n<div id=\"fs-id1165137895433\">\n<p id=\"fs-id1165135203531\">[latex]{a}_{n}=\\left\\{10,-3,0.9,-0.27,...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135181595\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135181595\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135181595\"]\n<p id=\"fs-id1165137785100\">[latex]\\begin{array}{cc}{a}_{1}=10,&amp; {a}_{n}=-0.3{a}_{n-1}\\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137595167\">\n<div id=\"fs-id1165137757979\">\n<p id=\"fs-id1165137757981\">[latex]{a}_{n}=\\left\\{0.61,1.83,5.49,16.47,...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137434224\">\n<div>\n<p id=\"fs-id1165137731174\">[latex]{a}_{n}=\\left\\{\\frac{3}{5},\\frac{1}{10},\\frac{1}{60},\\frac{1}{360},...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137749862\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137749862\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137749862\"]\n<p id=\"fs-id1165137726435\">[latex]\\begin{array}{cc}{a}_{1}=\\frac{3}{5},&amp; {a}_{n}=\\frac{1}{6}{a}_{n-1}\\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137784928\">\n<div id=\"fs-id1165137784930\">\n<p id=\"fs-id1165137447912\">[latex]{a}_{n}=\\left\\{-2,\\frac{4}{3},-\\frac{8}{9},\\frac{16}{27},...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135536510\">\n<div id=\"fs-id1165135536513\">\n<p id=\"fs-id1165137758236\">[latex]{a}_{n}=\\left\\{\\frac{1}{512},-\\frac{1}{128},\\frac{1}{32},-\\frac{1}{8},...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137827871\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137827871\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137827871\"]\n<p id=\"fs-id1165137595578\">[latex]{a}_{1}=\\frac{1}{512},{a}_{n}=-4{a}_{n-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137610897\">For the following exercises, write the first five terms of the geometric sequence.<\/p>\n\n<div id=\"fs-id1165137847172\">\n<div id=\"fs-id1165135185964\">\n<p id=\"fs-id1165135185966\">[latex]{a}_{n}=-4\\cdot {5}^{n-1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137725946\">\n<div id=\"fs-id1165137725948\">\n<p id=\"fs-id1165137651833\">[latex]{a}_{n}=12\\cdot {\\left(-\\frac{1}{2}\\right)}^{n-1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135161218\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135161218\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135161218\"]\n<p id=\"fs-id1165137653407\">[latex]12,-6,3,-\\frac{3}{2},\\frac{3}{4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135194312\">For the following exercises, write an explicit formula for each geometric sequence.<\/p>\n\n<div id=\"fs-id1165137425294\">\n<div id=\"fs-id1165137425296\">\n<p id=\"fs-id1165137549377\">[latex]{a}_{n}=\\left\\{-2,-4,-8,-16,...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137740800\">\n<div id=\"fs-id1165135397968\">\n<p id=\"fs-id1165135397970\">[latex]{a}_{n}=\\left\\{1,3,9,27,...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137531020\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137531020\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137531020\"]\n<p id=\"fs-id1165137531022\">[latex]{a}_{n}={3}^{n-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135177742\">\n<div id=\"fs-id1165135177745\">\n<p id=\"fs-id1165137760709\">[latex]{a}_{n}=\\left\\{-4,-12,-36,-108,...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737769\">\n<div id=\"fs-id1165137726013\">\n<p id=\"fs-id1165137726015\">[latex]{a}_{n}=\\left\\{0.8,-4,20,-100,...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165137446411\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137446411\"]\n<p id=\"fs-id1165137446411\">[latex]{a}_{n}=0.8\\cdot {\\left(-5\\right)}^{n-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137422465\">\n<div id=\"fs-id1165137451895\">\n<p id=\"fs-id1165137451897\">[latex]{a}_{n}=\\left\\{-1.25,-5,-20,-80,...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137666609\">\n<div id=\"fs-id1165137472730\">\n<p id=\"fs-id1165137472732\">[latex]{a}_{n}=\\left\\{-1,-\\frac{4}{5},-\\frac{16}{25},-\\frac{64}{125},...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137447709\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137447709\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137447709\"]\n<p id=\"fs-id1165137598008\">[latex]{a}_{n}=-{\\left(\\frac{4}{5}\\right)}^{n-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137565688\">\n<div id=\"fs-id1165137666683\">\n<p id=\"fs-id1165137666685\">[latex]{a}_{n}=\\left\\{2,\\frac{1}{3},\\frac{1}{18},\\frac{1}{108},...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137583712\">\n<div id=\"fs-id1165137583714\">\n<p id=\"fs-id1165137666916\">[latex]{a}_{n}=\\left\\{3,-1,\\frac{1}{3},-\\frac{1}{9},...\\right\\}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137779044\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137779044\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137779044\"]\n<p id=\"fs-id1165137779046\">[latex]{a}_{n}=3\\cdot {\\left(-\\frac{1}{3}\\right)}^{n-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137936625\">For the following exercises, find the specified term for the geometric sequence given.<\/p>\n\n<div id=\"fs-id1165137653204\">\n<div id=\"fs-id1165137653206\">\n<p id=\"fs-id1165135173345\">Let [latex]{a}_{1}=4,[\/latex] [latex]{a}_{n}=-3{a}_{n-1}.[\/latex] Find [latex]{a}_{8}.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191652\">\n<div id=\"fs-id1165135546049\">\n\nLet [latex]{a}_{n}=-{\\left(-\\frac{1}{3}\\right)}^{n-1}.[\/latex] Find [latex]{a}_{12}.[\/latex]\n\n<\/div>\n<div id=\"fs-id1165137793557\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137793557\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137793557\"]\n<p id=\"fs-id1165137793559\">[latex]{a}_{12}=\\frac{1}{177,147}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137871199\">For the following exercises, find the number of terms in the given finite geometric sequence.<\/p>\n\n<div id=\"fs-id1165137455508\">\n<div id=\"fs-id1165137455510\">\n<p id=\"fs-id1165137581440\">[latex]{a}_{n}=\\left\\{-1,3,-9,...,2187\\right\\}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137635269\">\n<div id=\"fs-id1165137417215\">\n<p id=\"fs-id1165137417217\">[latex]{a}_{n}=\\left\\{2,1,\\frac{1}{2},...,\\frac{1}{1024}\\right\\}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137675968\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137675968\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137675968\"]\n<p id=\"fs-id1165137675970\">There are [latex]12[\/latex] terms in the sequence.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137638147\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165135531584\">For the following exercises, determine whether the graph shown represents a geometric sequence.<\/p>\n\n<div id=\"fs-id1165137506987\">\n<div id=\"fs-id1165137898910\"><span id=\"fs-id1165137570178\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154901\/CNX_Precalc_Figure_11_03_201.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, -3), (2, -1), (3, 1), (4, 3), and (5, 5). The x-axis is labeled n and the y-axis is labeled a_n.\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137759706\">\n<div id=\"fs-id1165137748460\"><span id=\"fs-id1165137601995\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154903\/CNX_Precalc_Figure_11_03_202.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, -0.5), (2, 0.25), (3, 1.375), (4, 3.0625), and (5, 5.5938). The x-axis is labeled n and the y-axis is labeled a_n.\"><\/span><\/div>\n<div id=\"fs-id1165137663534\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137663534\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137663534\"]\n<p id=\"fs-id1165137663536\">The graph does not represent a geometric sequence.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\nFor the following exercises, use the information provided to graph the first five terms of the geometric sequence.\n<div id=\"fs-id1165137445372\">\n<div id=\"fs-id1165137612112\">\n<p id=\"fs-id1165137612114\">[latex]\\begin{array}{cc}{a}_{1}=1,&amp; r=\\frac{1}{2}\\end{array}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135309774\">\n<p id=\"fs-id1165137578258\">[latex]\\begin{array}{cc}{a}_{1}=3,&amp; {a}_{n}=2{a}_{n-1}\\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137784836\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137784836\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137784836\"]<span id=\"fs-id1165137634212\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154907\/CNX_Precalc_Figure_11_03_203.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, 3), (2, 6), (3, 12), (4, 24), and (5, 48). The x-axis is labeled n and the y-axis is labeled a_n.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137824382\">\n<div id=\"fs-id1165137659846\">\n<p id=\"fs-id1165137659848\">[latex]{a}_{n}=27\\cdot {0.3}^{n-1}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137611626\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165137705671\">\n<div id=\"fs-id1165135195117\">\n<p id=\"fs-id1165135195119\">Use recursive formulas to give two examples of geometric sequences whose 3<sup>rd<\/sup> terms are[latex]\\,200.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135191920\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135191920\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135191920\"]\n<p id=\"fs-id1165135191922\">Answers will vary. Examples: [latex]{\\begin{array}{cc}{a}_{1}=800,&amp; {a}_{n}=0.5a\\end{array}}_{n-1}[\/latex] and [latex]{\\begin{array}{cc}{a}_{1}=12.5,&amp; {a}_{n}=4a\\end{array}}_{n-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137706990\">\n<div id=\"fs-id1165137431466\">\n<p id=\"fs-id1165137431468\">Use explicit formulas to give two examples of geometric sequences whose 7<sup>th<\/sup> terms are [latex]1024.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137847161\">\n<div id=\"fs-id1165137399742\">\n<p id=\"fs-id1165137399744\">Find the 5<sup>th<\/sup> term of the geometric sequence [latex]\\left\\{b,4b,16b,...\\right\\}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137419654\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137419654\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137419654\"]\n<p id=\"fs-id1165137419656\">[latex]{a}_{5}=256b[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705775\">\n<div id=\"fs-id1165137705777\">\n<p id=\"fs-id1165137570885\">Find the 7<sup>th<\/sup> term of the geometric sequence [latex]\\left\\{64a\\left(-b\\right),32a\\left(-3b\\right),16a\\left(-9b\\right),...\\right\\}.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137823109\">\n<div id=\"fs-id1165137823111\">\n<p id=\"fs-id1165135194604\">At which term does the sequence [latex]\\left\\{10,12,14.4,17.28,\\text{ }...\\right\\}[\/latex] exceed [latex]100?[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137439223\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137439223\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137439223\"]\n<p id=\"fs-id1165137439224\">The sequence exceeds [latex]100[\/latex] at the 14<sup>th<\/sup> term, [latex]{a}_{14}\\approx 107.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135208918\">\n<div id=\"fs-id1165137823336\">\n<p id=\"fs-id1165137823338\">At which term does the sequence [latex]\\left\\{\\frac{1}{2187},\\frac{1}{729},\\frac{1}{243},\\frac{1}{81}\\text{ }...\\right\\}[\/latex] begin to have integer values?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137445687\">\n<div id=\"fs-id1165137675569\">\n<p id=\"fs-id1165137675572\">For which term does the geometric sequence [latex]{a}_{{}_{n}}=-36{\\left(\\frac{2}{3}\\right)}^{n-1}[\/latex] first have a non-integer value?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137660698\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137660698\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137660698\"]\n<p id=\"fs-id1165135377151\">[latex]{a}_{4}=-\\frac{32}{3}\\,[\/latex]is the first non-integer value<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137732788\">\n<div id=\"fs-id1165137732790\">\n\nUse the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the 10<sup>th<\/sup> term.\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137400201\">\n<div id=\"fs-id1165137400203\">\n<p id=\"fs-id1165137809947\">Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. Show the first 4 terms, and then find the 8<sup>th<\/sup> term.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137666622\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137666622\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137666622\"]\n<p id=\"fs-id1165137666624\">Answers will vary. Example: Explicit formula with a decimal common ratio: [latex]{a}_{n}=400\\cdot {0.5}^{n-1};[\/latex] First 4 terms: [latex]\\begin{array}{cc}400,200,100,50;&amp; {a}_{8}=3.125\\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135187388\">\n<div id=\"fs-id1165135187390\">\n<p id=\"fs-id1165135187392\">Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137740810\">\n \t<dt>common ratio<\/dt>\n \t<dd id=\"fs-id1165137849293\">the ratio between any two consecutive terms in a geometric sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137611024\">\n \t<dt>geometric sequence<\/dt>\n \t<dd id=\"fs-id1165137673421\">a sequence in which the ratio of a term to a previous term is a constant<\/dd>\n<\/dl>\n<\/div>\n\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Find the common ratio for a geometric sequence.<\/li>\n<li>List the terms of a geometric sequence.<\/li>\n<li>Use a recursive formula for a geometric sequence.<\/li>\n<li>Use an explicit formula for a geometric sequence.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137639564\">Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section, we will review sequences that grow in this way.<\/p>\n<div id=\"fs-id1165137715137\" class=\"bc-section section\">\n<h3>Finding Common Ratios<\/h3>\n<p id=\"fs-id1165137894517\">The yearly salary values described form a <strong>geometric sequence<\/strong> because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the <strong>common ratio<\/strong>. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.<\/p>\n<p><span id=\"fs-id1165137730802\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154844\/CNX_Precalc_Figure_11_03_001.jpg\" alt=\"A sequence , {1, 6, 36, 216, 1296, ...} that shows all the numbers have a common ratio of 6.\" \/><\/span><\/p>\n<div id=\"fs-id1165137749987\" class=\"textbox key-takeaways\">\n<h3>Definition of a Geometric Sequence<\/h3>\n<p id=\"fs-id1165137855264\">A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If [latex]{a}_{1}[\/latex] is the initial term of a geometric sequence and [latex]r[\/latex] is the common ratio, the sequence will be<\/p>\n<div id=\"eip-246\" class=\"unnumbered aligncenter\">[latex]\\left\\{{a}_{1}, \\,{a}_{1}r,\\,{a}_{1}{r}^{2},\\,{a}_{1}{r}^{3},\\,...\\right\\}.[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137417317\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135388542\"><strong>Given a set of numbers, determine if they represent a geometric sequence.<\/strong><\/p>\n<ol id=\"fs-id1165137399185\" type=\"1\">\n<li>Divide each term by the previous term.<\/li>\n<li>Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_03_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137594043\">\n<div id=\"fs-id1165137736734\">\n<h3>Finding Common Ratios<\/h3>\n<p id=\"fs-id1165137462087\">Is the sequence geometric? If so, find the common ratio.<\/p>\n<ol id=\"fs-id1165137911197\" type=\"a\">\n<li>[latex]1\\text{,}\\,2\\text{,}\\,4\\text{,}\\,8\\text{,}\\,16\\text{,}\\,...[\/latex]<\/li>\n<li>[latex]48\\text{,}\\,12\\text{,}\\,4\\text{, }2\\text{,}\\,...[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137430766\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137430766&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137430766&#8243;]Divide each term by the previous term to determine whether a common ratio exists.<\/p>\n<ol id=\"fs-id1165137410224\" type=\"a\">\n<li>[latex]\\begin{array}{llllllllll}\\frac{2}{1}=2\\hfill & \\hfill & \\hfill & \\frac{4}{2}=2\\hfill & \\hfill & \\hfill & \\frac{8}{4}=2\\hfill & \\hfill & \\hfill & \\frac{16}{8}=2\\hfill \\end{array}[\/latex]\n<p id=\"fs-id1165137837038\">The sequence is geometric because there is a common ratio. The common ratio is 2.<\/p>\n<\/li>\n<li>[latex]\\begin{array}{lllllll}\\frac{12}{48}=\\frac{1}{4}\\hfill & \\hfill & \\hfill & \\frac{4}{12}=\\frac{1}{3}\\hfill & \\hfill & \\hfill & \\frac{2}{4}=\\frac{1}{2}\\hfill \\end{array}[\/latex]\n<p id=\"fs-id1165137603152\">The sequence is not geometric because there is not a common ratio.<\/p>\n<\/li>\n<\/ol>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<div id=\"fs-id1165137667273\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137448166\">The graph of each sequence is shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_03_002\">(Figure)<\/a>. It seems from the graphs that both (a) and (b) appear have the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this interpretation holds, but (b) is not.<\/p>\n<div id=\"CNX_Precalc_Figure_11_03_002\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154847\/CNX_Precalc_Figure_11_03_002.jpg\" alt=\"Graph of two sequences where graph (a) is geometric and graph (b) is exponential.\" width=\"975\" height=\"286\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137742586\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137611774\"><strong>If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the common ratio?<\/strong><\/p>\n<p id=\"fs-id1165137803401\"><em>No. If you know that the sequence is geometric, you can choose any one term in the sequence and divide it by the previous term to find the common ratio.<\/em><\/p>\n<\/div>\n<div id=\"fs-id1165135241213\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_01\">\n<div id=\"fs-id1165137506295\">\n<p id=\"fs-id1165135408468\">Is the sequence geometric? If so, find the common ratio.<\/p>\n<div id=\"eip-id1491662\" class=\"unnumbered\">[latex]5,10,15,20,...[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137783622\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137783622&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137783622&#8243;]<\/p>\n<p id=\"fs-id1165137667314\">The sequence is not geometric because [latex]\\frac{10}{5}\\ne \\frac{15}{10}[\/latex].<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"eip-234\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_02\">\n<div id=\"fs-id1165137645227\">\n<p id=\"fs-id1165137462478\">Is the sequence geometric? If so, find the common ratio.<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=&#8221;415842&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;415842&#8243;]<\/p>\n<div>\n<div id=\"eip-id2227162\" class=\"unnumbered\">[latex]100,20,4,\\frac{4}{5},...[\/latex]<\/div>\n<\/div>\n<div>\n<p>The sequence is geometric. The common ratio is [latex]\\frac{1}{5}[\/latex].<\/p>\n<\/div>\n<p>[\/hidden-answer]<\/p><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137433726\" class=\"bc-section section\">\n<h3>Writing Terms of Geometric Sequences<\/h3>\n<p>Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[\/latex] and the common ratio is [latex]r=4,[\/latex] we can find subsequent terms by multiplying [latex]-2\\cdot 4[\/latex] to get [latex]-8[\/latex] then multiplying the result [latex]-8\\cdot 4[\/latex] to get [latex]-32[\/latex] and so on.<\/p>\n<div id=\"eip-245\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{a}_{1}=-2\\hfill \\\\ {a}_{2}=\\left(-2\\cdot 4\\right)=-8\\hfill \\\\ {a}_{3}=\\left(-8\\cdot 4\\right)=-32\\hfill \\\\ {a}_{4}=\\left(-32\\cdot 4\\right)=-128\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137472075\">The first four terms are[latex]\\left\\{\u20132\\text{, }\u20138\\text{, }\u201332\\text{, }\u2013128\\right\\}.[\/latex]<\/p>\n<div id=\"fs-id1165137939702\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137473764\"><strong>Given the first term and the common factor, find the first four terms of a geometric sequence.<\/strong><\/p>\n<ol id=\"fs-id1165137409884\" type=\"1\">\n<li>Multiply the initial term, [latex]{a}_{1},[\/latex] by the common ratio to find the next term, [latex]{a}_{2}.[\/latex]<\/li>\n<li>Repeat the process, using [latex]{a}_{n}={a}_{2}[\/latex] to find [latex]{a}_{3}[\/latex] and then [latex]{a}_{3}[\/latex] to find [latex]{a}_{4,}[\/latex] until all four terms have been identified.<\/li>\n<li>Write the terms separated by commons within brackets.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_03_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137573089\">\n<div id=\"fs-id1165137462497\">\n<h3>Writing the Terms of a Geometric Sequence<\/h3>\n<p id=\"fs-id1165137640196\">List the first four terms of the geometric sequence with [latex]{a}_{1}=5[\/latex] and [latex]r=\u20132.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137758063\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137758063&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137758063&#8243;]<\/p>\n<p id=\"fs-id1165137811084\">Multiply [latex]{a}_{1}[\/latex] by [latex]-2[\/latex] to find [latex]{a}_{2}.[\/latex] Repeat the process, using [latex]{a}_{2}[\/latex] to find [latex]{a}_{3},[\/latex]<br \/>\nand so on.<\/p>\n<div id=\"eip-id1165135160893\" class=\"unnumbered\">[latex]\\begin{array}{l}{a}_{1}=5\\hfill \\\\ {a}_{2}=-2{a}_{1}=-10\\hfill \\\\ {a}_{3}=-2{a}_{2}=20\\hfill \\\\ {a}_{4}=-2{a}_{3}=-40\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137560200\">The first four terms are [latex]\\left\\{5,\u201310,20,\u201340\\right\\}.[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137416531\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_03\">\n<div id=\"fs-id1165137619295\">\n<p>List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\frac{1}{3}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137655223\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137655223&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137655223&#8243;]<\/p>\n<p id=\"fs-id1165137560204\">[latex]\\left\\{18,6,2,\\frac{2}{3},\\frac{2}{9}\\right\\}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135694573\" class=\"bc-section section\">\n<h3>Using Recursive Formulas for Geometric Sequences<\/h3>\n<p id=\"fs-id1165137641691\">A <span class=\"no-emphasis\">recursive formula<\/span> allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.<\/p>\n<div id=\"fs-id1165137448332\" class=\"textbox key-takeaways\">\n<h3>Recursive Formula for a Geometric Sequence<\/h3>\n<p id=\"fs-id1165137463184\">The recursive formula for a geometric sequence with common ratio [latex]r[\/latex] and first term [latex]{a}_{1}[\/latex] is<\/p>\n<div id=\"fs-id1165137662634\">[latex]{a}_{n}=r{a}_{n-1},n\\ge 2[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135207482\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137559120\"><strong>Given the first several terms of a geometric sequence, write its recursive formula.<\/strong><\/p>\n<ol id=\"fs-id1165137442323\" type=\"1\">\n<li>State the initial term.<\/li>\n<li>Find the common ratio by dividing any term by the preceding term.<\/li>\n<li>Substitute the common ratio into the recursive formula for a geometric sequence.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_03_03\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165137640113\">\n<h3>Using Recursive Formulas for Geometric Sequences<\/h3>\n<p id=\"fs-id1165137470342\">Write a recursive formula for the following geometric sequence.<\/p>\n<div id=\"eip-id1165132059630\" class=\"unnumbered\">[latex]\\left\\{6\\text{, }9\\text{, }13.5\\text{, }20.25\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135190538\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165135190538&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165135190538&#8243;]<\/p>\n<p id=\"fs-id1165137645992\">The first term is given as 6. The common ratio can be found by dividing the second term by the first term.<\/p>\n<div id=\"eip-id1165134410969\" class=\"unnumbered\">[latex]r=\\frac{9}{6}=1.5[\/latex]<\/div>\n<p id=\"fs-id1165137539505\">Substitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}.[\/latex]<\/p>\n<div id=\"eip-id1165135694281\" class=\"unnumbered\">[latex]\\begin{array}{l}{a}_{n}=r{a}_{n-1}\\\\ {a}_{n}=1.5{a}_{n-1}\\text{ for }n\\ge 2\\\\ {a}_{1}=6\\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137629524\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137409471\">The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_03_003\">(Figure)<\/a><\/p>\n<div id=\"CNX_Precalc_Figure_11_03_003\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154849\/CNX_Precalc_Figure_11_03_003.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"215\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137804164\" class=\"textbox key-takeaways\">\n<p id=\"fs-id1165137594708\"><strong>Do we have to divide the second term by the first term to find the common ratio?<\/strong><\/p>\n<p id=\"fs-id1165135593534\"><em>No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.<\/em><\/p>\n<\/div>\n<div id=\"fs-id1165137461314\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_05\">\n<div id=\"fs-id1165137538955\">\n<p id=\"fs-id1165137418734\">Write a recursive formula for the following geometric sequence.<\/p>\n<div id=\"eip-id1264736\" class=\"unnumbered\">[latex]\\left\\{2\\text{, }\\frac{4}{3}\\text{, }\\frac{8}{9}\\text{, }\\frac{16}{27}\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137605332\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137605332&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137605332&#8243;]<\/p>\n<p id=\"fs-id1165137779221\">[latex]\\begin{array}{l}{a}_{1}=2\\\\ {a}_{n}=\\frac{2}{3}{a}_{n-1}\\text{ for }n\\ge 2\\end{array}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137656805\" class=\"bc-section section\">\n<h3>Using Explicit Formulas for Geometric Sequences<\/h3>\n<p id=\"fs-id1165137507452\">Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.<\/p>\n<div class=\"unnumbered\">[latex]{a}_{n}={a}_{1}{r}^{n-1}[\/latex]<\/div>\n<p id=\"fs-id1165137410536\">Let\u2019s take a look at the sequence [latex]\\left\\{18\\text{, }36\\text{, }72\\text{, }144\\text{, }288\\text{, }...\\right\\}.[\/latex] This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is<\/p>\n<div id=\"eip-422\" class=\"unnumbered aligncenter\">[latex]{a}_{n}=18\u00b7{2}^{n-1}[\/latex]<\/div>\n<p id=\"fs-id1165137473099\">The graph of the sequence is shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_03_004\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_11_03_004\" class=\"small aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154852\/CNX_Precalc_Figure_11_03_004.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"440\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165137647200\" class=\"textbox key-takeaways\">\n<h3>Explicit Formula for a Geometric Sequence<\/h3>\n<p id=\"fs-id1165137835426\">The <em>n<\/em>th term of a geometric sequence is given by the <span class=\"no-emphasis\">explicit formula<\/span>:<\/p>\n<div id=\"fs-id1165137461602\">[latex]{a}_{n}={a}_{1}{r}^{n-1}[\/latex]<\/div>\n<\/div>\n<div id=\"Example_11_03_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137453521\">\n<div id=\"fs-id1165137811825\">\n<h3>Writing Terms of Geometric Sequences Using the Explicit Formula<\/h3>\n<p id=\"fs-id1165137874642\">Given a geometric sequence with[latex]\\,{a}_{1}=3\\,[\/latex]and[latex]\\,{a}_{4}=24,\\,[\/latex]find [latex]{a}_{2}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137408599\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137408599&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137408599&#8243;]The sequence can be written in terms of the initial term and the common ratio[latex]\\,r.[\/latex]<\/p>\n<div id=\"eip-id1165134060418\" class=\"unnumbered\">[latex]3,3r,3{r}^{2},3{r}^{3},...[\/latex]<\/div>\n<p id=\"fs-id1165137432346\">Find the common ratio using the given fourth term.<\/p>\n<div id=\"eip-id1165137444653\" class=\"unnumbered\">[latex]\\begin{array}{ll}{a}_{n}={a}_{1}{r}^{n-1}\\hfill & \\hfill \\\\ {a}_{4}=3{r}^{3}\\hfill & \\text{Write the fourth term of sequence in terms of }{\\alpha }_{1}\\,\\text{and }r\\hfill \\\\ 24=3{r}^{3}\\hfill & \\text{Substitute }24\\text{ for}\\,{a}_{4}\\hfill \\\\ \\,\\,\\,8={r}^{3}\\hfill & \\text{Divide}\\hfill \\\\ \\,\\,\\,r=2\\hfill & \\text{Solve for the common ratio}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137455556\">Find the second term by multiplying the first term by the common ratio.<\/p>\n<div id=\"eip-id1165135426318\" class=\"unnumbered\">[latex]\\begin{array}{ll}{a}_{2}\\hfill & =2{a}_{1}\\hfill \\\\ \\hfill & =2\\left(3\\right)\\hfill \\\\ \\hfill & =6\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137725528\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137789060\">The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137444993\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_04\">\n<div id=\"fs-id1165137459454\">\n<p id=\"fs-id1165137653766\">Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex], find [latex]{a}_{6}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137447458\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137447458&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137447458&#8243;]<\/p>\n<p id=\"fs-id1165137532391\">[latex]{a}_{6}=16,384[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_11_03_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137611787\">\n<div id=\"fs-id1165135255552\">\n<h3>Writing an Explicit Formula for the <em>n<\/em>th Term of a Geometric Sequence<\/h3>\n<p id=\"fs-id1165137414859\">Write an explicit formula for the [latex]n\\text{th}[\/latex] term of the following geometric sequence.<\/p>\n<div id=\"eip-id1165134113698\" class=\"unnumbered\">[latex]\\left\\{2\\text{, }10\\text{, }50\\text{, }250\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137911060\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137911060&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137911060&#8243;]The first term is 2. The common ratio can be found by dividing the second term by the first term.<\/p>\n<div id=\"eip-id1165134085791\" class=\"unnumbered\">[latex]\\frac{10}{2}=5[\/latex]<\/div>\n<p id=\"fs-id1165137407052\">The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.<\/p>\n<div id=\"eip-id1165137558360\" class=\"unnumbered\">[latex]\\begin{array}{l}{a}_{n}={a}_{1}{r}^{\\left(n-1\\right)}\\hfill \\\\ {a}_{n}=2\\cdot {5}^{n-1}\\hfill \\end{array}[\/latex]<\/div>\n<p>The graph of this sequence in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_03_005\">(Figure)<\/a> shows an exponential pattern.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154854\/CNX_Precalc_Figure_11_03_005.jpg\" alt=\"Graph of the geometric sequence.\" width=\"487\" height=\"290\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165137841605\">[\/hidden-answer]<span id=\"fs-id1165137460332\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137600603\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_06\">\n<div id=\"fs-id1165137762502\">\n<p id=\"fs-id1165137528367\">Write an explicit formula for the following geometric sequence.<\/p>\n<div id=\"eip-id1673413\" class=\"unnumbered\">[latex]\\left\\{\u20131\\text{, }3\\text{, }\u20139\\text{, }27\\text{, }...\\right\\}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137532482\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137532482&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137532482&#8243;]<\/p>\n<p id=\"fs-id1165137435217\">[latex]{a}_{n}=-{\\left(-3\\right)}^{n-1}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135251448\" class=\"bc-section section\">\n<h3>Solving Application Problems with Geometric Sequences<\/h3>\n<p id=\"fs-id1165137436841\">In real-world scenarios involving arithmetic sequences, we may need to use an initial term of [latex]{a}_{0}[\/latex] instead of [latex]{a}_{1}.\\,[\/latex]In these problems, we can alter the explicit formula slightly by using the following formula:<\/p>\n<div id=\"fs-id1165137455099\" class=\"unnumbered aligncenter\">[latex]{a}_{n}={a}_{0}{r}^{n}[\/latex]<\/div>\n<div id=\"Example_11_03_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137675504\">\n<div id=\"fs-id1165135207496\">\n<h3>Solving Application Problems with Geometric Sequences<\/h3>\n<p id=\"fs-id1165137724701\">In 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.<\/p>\n<ol id=\"fs-id1165137724573\" type=\"a\">\n<li>Write a formula for the student population.<\/li>\n<li>Estimate the student population in 2020.<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137603324\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137603324&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137603324&#8243;]<\/p>\n<ol id=\"fs-id1165137812960\" type=\"a\">\n<li>\n<p id=\"fs-id1165137475561\">The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04.<\/p>\n<p id=\"fs-id1165135345013\">Let [latex]P[\/latex] be the student population and [latex]n[\/latex] be the number of years after 2013. Using the explicit formula for a geometric sequence we get<\/p>\n<div id=\"eip-id1165134384459\" class=\"unnumbered\">[latex]{P}_{n} =284\\cdot {1.04}^{n}[\/latex]<\/div>\n<\/li>\n<li>\n<p id=\"fs-id1165137472563\">We can find the number of years since 2013 by subtracting.<\/p>\n<div id=\"eip-id1165137664601\" class=\"unnumbered\">[latex]2020-2013=7[\/latex]<\/div>\n<p id=\"fs-id1165137453691\">We are looking for the population after 7 years. We can substitute 7 for [latex]n[\/latex] to estimate the population in 2020.<\/p>\n<div id=\"eip-id1165135699078\" class=\"unnumbered\">[latex]{P}_{7}=284\\cdot {1.04}^{7}\\approx 374[\/latex]<\/div>\n<p id=\"fs-id1165137528682\">The student population will be about 374 in 2020.[\/hidden-answer]<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137768542\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_03_07\">\n<div id=\"fs-id1165137410731\">\n<p id=\"fs-id1165137834028\">A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.<\/p>\n<ol id=\"eip-id1165134294794\" type=\"a\">\n<li>Write a formula for the number of hits.<\/li>\n<li>Estimate the number of hits in 5 weeks.<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137455278\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137455278&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137455278&#8243;]<\/p>\n<ol id=\"eip-id1165135538601\" type=\"a\">\n<li>[latex]{P}_{n} = 293\\cdot 1.026{a}^{n}[\/latex]<\/li>\n<li>The number of hits will be about 333.<\/li>\n<\/ol>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137443318\" class=\"precalculus media\">\n<p id=\"fs-id1165137817563\">Access these online resources for additional instruction and practice with geometric sequences.<\/p>\n<ul>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/geometricseq\">Geometric Sequences<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/sequencetype\">Determine the Type of Sequence<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/sequenceformula\">Find the Formula for a Sequence<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135264797\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165133155748\" summary=\"..\">\n<tbody>\n<tr>\n<td>recursive formula for [latex]nth[\/latex] term of a geometric sequence<\/td>\n<td>[latex]{a}_{n}=r{a}_{n-1},n\\ge 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>explicit formula for[latex]\\,nth\\,[\/latex]term of a geometric sequence<\/td>\n<td>[latex]{a}_{n}={a}_{1}{r}^{n-1}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137611739\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135189853\">\n<li>A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.<\/li>\n<li>The constant ratio between two consecutive terms is called the common ratio.<\/li>\n<li>The common ratio can be found by dividing any term in the sequence by the previous term. See <a class=\"autogenerated-content\" href=\"#Example_11_03_01\">(Figure)<\/a>.<\/li>\n<li>The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See <a class=\"autogenerated-content\" href=\"#Example_11_03_02\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_11_03_04\">(Figure)<\/a>.<\/li>\n<li>A recursive formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]\\,{a}_{n}=r{a}_{n\u20131}\\,[\/latex]for [latex]n\\ge 2[\/latex].<\/li>\n<li>As with any recursive formula, the initial term of the sequence must be given. See <a class=\"autogenerated-content\" href=\"#Example_11_03_03\">(Figure)<\/a>.<\/li>\n<li>An explicit formula for a geometric sequence with common ratio [latex]r[\/latex] is given by [latex]\\,{a}_{n}={a}_{1}{r}^{n\u20131}.[\/latex] See <a class=\"autogenerated-content\" href=\"#Example_11_03_05\">(Figure)<\/a>.<\/li>\n<li>In application problems, we sometimes alter the explicit formula slightly to [latex]\\,{a}_{n}={a}_{0}{r}^{n}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_11_03_06\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137695322\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137603819\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137415276\">\n<div id=\"fs-id1165137601852\">\n<p id=\"fs-id1165137460182\">What is a geometric sequence?<\/p>\n<\/div>\n<div id=\"fs-id1165137413921\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137413921&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137413921&#8243;]<\/p>\n<p id=\"fs-id1165137419043\">A sequence in which the ratio between any two consecutive terms is constant.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135386446\">\n<div id=\"fs-id1165137430484\">\n<p id=\"fs-id1165137430485\">How is the common ratio of a geometric sequence found?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137651855\">\n<div id=\"fs-id1165137871059\">\n<p id=\"fs-id1165137666549\">What is the procedure for determining whether a sequence is geometric?<\/p>\n<\/div>\n<div id=\"fs-id1165137571584\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137571584&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137571584&#8243;]<\/p>\n<p id=\"fs-id1165137645826\">Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137786269\">\n<div id=\"fs-id1165137698332\">\n<p id=\"fs-id1165137767192\">What is the difference between an arithmetic sequence and a geometric sequence?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137806166\">\n<div id=\"fs-id1165137812741\">\n<p>Describe how exponential functions and geometric sequences are similar. How are they different?<\/p>\n<\/div>\n<div id=\"fs-id1165137410439\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137410439&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137410439&#8243;]<\/p>\n<p id=\"fs-id1165137735983\">Both geometric sequences and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an exponential function must be positive.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135185333\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p>For the following exercises, find the common ratio for the geometric sequence.<\/p>\n<div id=\"fs-id1165137656402\">\n<div id=\"fs-id1165137628207\">\n<p id=\"fs-id1165137643000\">[latex]1,3,9,27,81,...[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137698185\">\n<div id=\"fs-id1165137433830\">\n<p id=\"fs-id1165137936860\">[latex]-0.125,0.25,-0.5,1,-2,...[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137560236\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137560236&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137560236&#8243;]<\/p>\n<p id=\"fs-id1165137626648\">The common ratio is [latex]-2[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137696255\">\n<div id=\"fs-id1165137666722\">\n<p id=\"fs-id1165137925247\">[latex]-2,-\\frac{1}{2},-\\frac{1}{8},-\\frac{1}{32},-\\frac{1}{128},...[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.<\/p>\n<div id=\"fs-id1165137533956\">\n<div id=\"fs-id1165137643508\">\n<p id=\"fs-id1165137725448\">[latex]-6,-12,-24,-48,-96,...[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137660469\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137660469&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137660469&#8243;]<\/p>\n<p id=\"fs-id1165137550725\">The sequence is geometric. The common ratio is 2.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137462907\">\n<div id=\"fs-id1165137452609\">\n<p id=\"fs-id1165137803618\">[latex]5,5.2,5.4,5.6,5.8,...[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137823090\">\n<div id=\"fs-id1165137433632\">\n<p id=\"fs-id1165137823320\">[latex]-1,\\frac{1}{2},-\\frac{1}{4},\\frac{1}{8},-\\frac{1}{16},...[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134047692\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165134047692&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165134047692&#8243;]<\/p>\n<p id=\"fs-id1165137911673\">The sequence is geometric. The common ratio is [latex]-\\frac{1}{2}.[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137935634\">\n<div id=\"fs-id1165137517318\">\n<p id=\"fs-id1165137723782\">[latex]6,8,11,15,20,...[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137841659\">\n<div id=\"fs-id1165137841662\">\n<p id=\"fs-id1165137542166\">[latex]0.8,4,20,100,500,...[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137849116\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137849116&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137849116&#8243;]<\/p>\n<p id=\"fs-id1165137849119\">The sequence is geometric. The common ratio is [latex]5.[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137578676\">For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.<\/p>\n<div id=\"fs-id1165135175313\">\n<div id=\"fs-id1165137566099\">\n<p id=\"fs-id1165137566101\">[latex]\\begin{array}{cc}{a}_{1}=8,& r=0.3\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447376\">\n<div id=\"fs-id1165137539733\">\n<p id=\"fs-id1165135172312\">[latex]\\begin{array}{cc}{a}_{1}=5,& r=\\frac{1}{5}\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137444326\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137444326&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137444326&#8243;]<\/p>\n<p id=\"fs-id1165135149206\">[latex]5,1,\\frac{1}{5},\\frac{1}{25},\\frac{1}{125}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135152167\">For the following exercises, write the first five terms of the geometric sequence, given any two terms.<\/p>\n<div id=\"fs-id1165135195397\">\n<div id=\"fs-id1165137898879\">\n<p id=\"fs-id1165135666802\">[latex]\\begin{array}{cc}{a}_{7}=64,& {a}_{10}\\end{array}=512[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137694069\">\n<p id=\"fs-id1165137806803\">[latex]\\begin{array}{cc}{a}_{6}=25,& {a}_{8}\\end{array}=6.25[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137405237\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137405237&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137405237&#8243;]<\/p>\n<p id=\"fs-id1165137405239\">[latex]800,400,200,100,50[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137889906\">For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.<\/p>\n<div id=\"fs-id1165137635538\">\n<div id=\"fs-id1165137723274\">\n<p id=\"fs-id1165137529044\">The first term is [latex]2,[\/latex] and the common ratio is [latex]3.[\/latex] Find the 5<sup>th<\/sup> term.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193320\">\n<div id=\"fs-id1165137557071\">\n<p id=\"fs-id1165137602820\">The first term is 16 and the common ratio is [latex]-\\frac{1}{3}.[\/latex] Find the 4<sup>th<\/sup> term.<\/p>\n<\/div>\n<div id=\"fs-id1165135173567\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165135173567&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165135173567&#8243;]<\/p>\n<p id=\"fs-id1165135173569\">[latex]{a}_{4}=-\\frac{16}{27}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137460391\">For the following exercises, find the specified term for the geometric sequence, given the first four terms.<\/p>\n<div id=\"fs-id1165137415713\">\n<div id=\"fs-id1165137772362\">\n<p id=\"fs-id1165135174888\">[latex]{a}_{n}=\\left\\{-1,2,-4,8,...\\right\\}.[\/latex] Find [latex]{a}_{12}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137727503\">\n<div id=\"fs-id1165135191604\">\n<p id=\"fs-id1165135187574\">[latex]{a}_{n}=\\left\\{-2,\\frac{2}{3},-\\frac{2}{9},\\frac{2}{27},...\\right\\}.[\/latex] Find [latex]{a}_{7}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137450967\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137450967&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137450967&#8243;]<\/p>\n<p id=\"fs-id1165137450969\">[latex]{a}_{7}=-\\frac{2}{729}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137642161\">For the following exercises, write the first five terms of the geometric sequence.<\/p>\n<div id=\"fs-id1165137849215\">\n<div id=\"fs-id1165137811176\">\n<p id=\"fs-id1165137476786\">[latex]\\begin{array}{cc}{a}_{1}=-486,& {a}_{n}=-\\frac{1}{3}\\end{array}{a}_{n-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135195370\">\n<div id=\"fs-id1165135195372\">\n<p id=\"fs-id1165134108412\">[latex]\\begin{array}{cc}{a}_{1}=7,& {a}_{n}=0.2{a}_{n-1}\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137922652\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137922652&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137922652&#8243;]<\/p>\n<p id=\"fs-id1165135381368\">[latex]7,1.4,0.28,0.056,0.0112[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137698447\">For the following exercises, write a recursive formula for each geometric sequence.<\/p>\n<div id=\"fs-id1165137771764\">\n<div id=\"fs-id1165135496654\">\n<p id=\"fs-id1165135496656\">[latex]{a}_{n}=\\left\\{-1,5,-25,125,...\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137679367\">\n<div id=\"fs-id1165137679369\">\n<p id=\"fs-id1165137599938\">[latex]{a}_{n}=\\left\\{-32,-16,-8,-4,...\\right\\}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137422756\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137422756&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137422756&#8243;]<\/p>\n<p id=\"fs-id1165137653539\">[latex]\\begin{array}{cc}a{}_{1}=-32,& {a}_{n}=\\frac{1}{2}{a}_{n-1}\\end{array}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447992\">\n<div id=\"fs-id1165135543366\">\n<p id=\"fs-id1165135543368\">[latex]{a}_{n}=\\left\\{14,56,224,896,...\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137895431\">\n<div id=\"fs-id1165137895433\">\n<p id=\"fs-id1165135203531\">[latex]{a}_{n}=\\left\\{10,-3,0.9,-0.27,...\\right\\}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135181595\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165135181595&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165135181595&#8243;]<\/p>\n<p id=\"fs-id1165137785100\">[latex]\\begin{array}{cc}{a}_{1}=10,& {a}_{n}=-0.3{a}_{n-1}\\end{array}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137595167\">\n<div id=\"fs-id1165137757979\">\n<p id=\"fs-id1165137757981\">[latex]{a}_{n}=\\left\\{0.61,1.83,5.49,16.47,...\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137434224\">\n<div>\n<p id=\"fs-id1165137731174\">[latex]{a}_{n}=\\left\\{\\frac{3}{5},\\frac{1}{10},\\frac{1}{60},\\frac{1}{360},...\\right\\}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137749862\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137749862&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137749862&#8243;]<\/p>\n<p id=\"fs-id1165137726435\">[latex]\\begin{array}{cc}{a}_{1}=\\frac{3}{5},& {a}_{n}=\\frac{1}{6}{a}_{n-1}\\end{array}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137784928\">\n<div id=\"fs-id1165137784930\">\n<p id=\"fs-id1165137447912\">[latex]{a}_{n}=\\left\\{-2,\\frac{4}{3},-\\frac{8}{9},\\frac{16}{27},...\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135536510\">\n<div id=\"fs-id1165135536513\">\n<p id=\"fs-id1165137758236\">[latex]{a}_{n}=\\left\\{\\frac{1}{512},-\\frac{1}{128},\\frac{1}{32},-\\frac{1}{8},...\\right\\}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137827871\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137827871&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137827871&#8243;]<\/p>\n<p id=\"fs-id1165137595578\">[latex]{a}_{1}=\\frac{1}{512},{a}_{n}=-4{a}_{n-1}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137610897\">For the following exercises, write the first five terms of the geometric sequence.<\/p>\n<div id=\"fs-id1165137847172\">\n<div id=\"fs-id1165135185964\">\n<p id=\"fs-id1165135185966\">[latex]{a}_{n}=-4\\cdot {5}^{n-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137725946\">\n<div id=\"fs-id1165137725948\">\n<p id=\"fs-id1165137651833\">[latex]{a}_{n}=12\\cdot {\\left(-\\frac{1}{2}\\right)}^{n-1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135161218\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165135161218&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165135161218&#8243;]<\/p>\n<p id=\"fs-id1165137653407\">[latex]12,-6,3,-\\frac{3}{2},\\frac{3}{4}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135194312\">For the following exercises, write an explicit formula for each geometric sequence.<\/p>\n<div id=\"fs-id1165137425294\">\n<div id=\"fs-id1165137425296\">\n<p id=\"fs-id1165137549377\">[latex]{a}_{n}=\\left\\{-2,-4,-8,-16,...\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137740800\">\n<div id=\"fs-id1165135397968\">\n<p id=\"fs-id1165135397970\">[latex]{a}_{n}=\\left\\{1,3,9,27,...\\right\\}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137531020\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137531020&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137531020&#8243;]<\/p>\n<p id=\"fs-id1165137531022\">[latex]{a}_{n}={3}^{n-1}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135177742\">\n<div id=\"fs-id1165135177745\">\n<p id=\"fs-id1165137760709\">[latex]{a}_{n}=\\left\\{-4,-12,-36,-108,...\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737769\">\n<div id=\"fs-id1165137726013\">\n<p id=\"fs-id1165137726015\">[latex]{a}_{n}=\\left\\{0.8,-4,20,-100,...\\right\\}[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=&#8221;fs-id1165137446411&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137446411&#8243;]<\/p>\n<p id=\"fs-id1165137446411\">[latex]{a}_{n}=0.8\\cdot {\\left(-5\\right)}^{n-1}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137422465\">\n<div id=\"fs-id1165137451895\">\n<p id=\"fs-id1165137451897\">[latex]{a}_{n}=\\left\\{-1.25,-5,-20,-80,...\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137666609\">\n<div id=\"fs-id1165137472730\">\n<p id=\"fs-id1165137472732\">[latex]{a}_{n}=\\left\\{-1,-\\frac{4}{5},-\\frac{16}{25},-\\frac{64}{125},...\\right\\}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137447709\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137447709&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137447709&#8243;]<\/p>\n<p id=\"fs-id1165137598008\">[latex]{a}_{n}=-{\\left(\\frac{4}{5}\\right)}^{n-1}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137565688\">\n<div id=\"fs-id1165137666683\">\n<p id=\"fs-id1165137666685\">[latex]{a}_{n}=\\left\\{2,\\frac{1}{3},\\frac{1}{18},\\frac{1}{108},...\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137583712\">\n<div id=\"fs-id1165137583714\">\n<p id=\"fs-id1165137666916\">[latex]{a}_{n}=\\left\\{3,-1,\\frac{1}{3},-\\frac{1}{9},...\\right\\}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137779044\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137779044&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137779044&#8243;]<\/p>\n<p id=\"fs-id1165137779046\">[latex]{a}_{n}=3\\cdot {\\left(-\\frac{1}{3}\\right)}^{n-1}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137936625\">For the following exercises, find the specified term for the geometric sequence given.<\/p>\n<div id=\"fs-id1165137653204\">\n<div id=\"fs-id1165137653206\">\n<p id=\"fs-id1165135173345\">Let [latex]{a}_{1}=4,[\/latex] [latex]{a}_{n}=-3{a}_{n-1}.[\/latex] Find [latex]{a}_{8}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191652\">\n<div id=\"fs-id1165135546049\">\n<p>Let [latex]{a}_{n}=-{\\left(-\\frac{1}{3}\\right)}^{n-1}.[\/latex] Find [latex]{a}_{12}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137793557\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137793557&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137793557&#8243;]<\/p>\n<p id=\"fs-id1165137793559\">[latex]{a}_{12}=\\frac{1}{177,147}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137871199\">For the following exercises, find the number of terms in the given finite geometric sequence.<\/p>\n<div id=\"fs-id1165137455508\">\n<div id=\"fs-id1165137455510\">\n<p id=\"fs-id1165137581440\">[latex]{a}_{n}=\\left\\{-1,3,-9,...,2187\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137635269\">\n<div id=\"fs-id1165137417215\">\n<p id=\"fs-id1165137417217\">[latex]{a}_{n}=\\left\\{2,1,\\frac{1}{2},...,\\frac{1}{1024}\\right\\}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137675968\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137675968&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137675968&#8243;]<\/p>\n<p id=\"fs-id1165137675970\">There are [latex]12[\/latex] terms in the sequence.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137638147\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165135531584\">For the following exercises, determine whether the graph shown represents a geometric sequence.<\/p>\n<div id=\"fs-id1165137506987\">\n<div id=\"fs-id1165137898910\"><span id=\"fs-id1165137570178\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154901\/CNX_Precalc_Figure_11_03_201.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, -3), (2, -1), (3, 1), (4, 3), and (5, 5). The x-axis is labeled n and the y-axis is labeled a_n.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137759706\">\n<div id=\"fs-id1165137748460\"><span id=\"fs-id1165137601995\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154903\/CNX_Precalc_Figure_11_03_202.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, -0.5), (2, 0.25), (3, 1.375), (4, 3.0625), and (5, 5.5938). The x-axis is labeled n and the y-axis is labeled a_n.\" \/><\/span><\/div>\n<div id=\"fs-id1165137663534\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137663534&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137663534&#8243;]<\/p>\n<p id=\"fs-id1165137663536\">The graph does not represent a geometric sequence.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<p>For the following exercises, use the information provided to graph the first five terms of the geometric sequence.<\/p>\n<div id=\"fs-id1165137445372\">\n<div id=\"fs-id1165137612112\">\n<p id=\"fs-id1165137612114\">[latex]\\begin{array}{cc}{a}_{1}=1,& r=\\frac{1}{2}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135309774\">\n<p id=\"fs-id1165137578258\">[latex]\\begin{array}{cc}{a}_{1}=3,& {a}_{n}=2{a}_{n-1}\\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137784836\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137784836&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137784836&#8243;]<span id=\"fs-id1165137634212\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19154907\/CNX_Precalc_Figure_11_03_203.jpg\" alt=\"Graph of a scattered plot with labeled points: (1, 3), (2, 6), (3, 12), (4, 24), and (5, 48). The x-axis is labeled n and the y-axis is labeled a_n.\" \/><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137824382\">\n<div id=\"fs-id1165137659846\">\n<p id=\"fs-id1165137659848\">[latex]{a}_{n}=27\\cdot {0.3}^{n-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137611626\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165137705671\">\n<div id=\"fs-id1165135195117\">\n<p id=\"fs-id1165135195119\">Use recursive formulas to give two examples of geometric sequences whose 3<sup>rd<\/sup> terms are[latex]\\,200.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135191920\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165135191920&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165135191920&#8243;]<\/p>\n<p id=\"fs-id1165135191922\">Answers will vary. Examples: [latex]{\\begin{array}{cc}{a}_{1}=800,& {a}_{n}=0.5a\\end{array}}_{n-1}[\/latex] and [latex]{\\begin{array}{cc}{a}_{1}=12.5,& {a}_{n}=4a\\end{array}}_{n-1}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137706990\">\n<div id=\"fs-id1165137431466\">\n<p id=\"fs-id1165137431468\">Use explicit formulas to give two examples of geometric sequences whose 7<sup>th<\/sup> terms are [latex]1024.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137847161\">\n<div id=\"fs-id1165137399742\">\n<p id=\"fs-id1165137399744\">Find the 5<sup>th<\/sup> term of the geometric sequence [latex]\\left\\{b,4b,16b,...\\right\\}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137419654\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137419654&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137419654&#8243;]<\/p>\n<p id=\"fs-id1165137419656\">[latex]{a}_{5}=256b[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705775\">\n<div id=\"fs-id1165137705777\">\n<p id=\"fs-id1165137570885\">Find the 7<sup>th<\/sup> term of the geometric sequence [latex]\\left\\{64a\\left(-b\\right),32a\\left(-3b\\right),16a\\left(-9b\\right),...\\right\\}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137823109\">\n<div id=\"fs-id1165137823111\">\n<p id=\"fs-id1165135194604\">At which term does the sequence [latex]\\left\\{10,12,14.4,17.28,\\text{ }...\\right\\}[\/latex] exceed [latex]100?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137439223\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137439223&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137439223&#8243;]<\/p>\n<p id=\"fs-id1165137439224\">The sequence exceeds [latex]100[\/latex] at the 14<sup>th<\/sup> term, [latex]{a}_{14}\\approx 107.[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135208918\">\n<div id=\"fs-id1165137823336\">\n<p id=\"fs-id1165137823338\">At which term does the sequence [latex]\\left\\{\\frac{1}{2187},\\frac{1}{729},\\frac{1}{243},\\frac{1}{81}\\text{ }...\\right\\}[\/latex] begin to have integer values?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137445687\">\n<div id=\"fs-id1165137675569\">\n<p id=\"fs-id1165137675572\">For which term does the geometric sequence [latex]{a}_{{}_{n}}=-36{\\left(\\frac{2}{3}\\right)}^{n-1}[\/latex] first have a non-integer value?<\/p>\n<\/div>\n<div id=\"fs-id1165137660698\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137660698&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137660698&#8243;]<\/p>\n<p id=\"fs-id1165135377151\">[latex]{a}_{4}=-\\frac{32}{3}\\,[\/latex]is the first non-integer value<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137732788\">\n<div id=\"fs-id1165137732790\">\n<p>Use the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the 10<sup>th<\/sup> term.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137400201\">\n<div id=\"fs-id1165137400203\">\n<p id=\"fs-id1165137809947\">Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. Show the first 4 terms, and then find the 8<sup>th<\/sup> term.<\/p>\n<\/div>\n<div id=\"fs-id1165137666622\" class=\"solution textbox shaded\">[reveal-answer q=&#8221;fs-id1165137666622&#8243;]Show Solution[\/reveal-answer]<br \/>\n[hidden-answer a=&#8221;fs-id1165137666622&#8243;]<\/p>\n<p id=\"fs-id1165137666624\">Answers will vary. Example: Explicit formula with a decimal common ratio: [latex]{a}_{n}=400\\cdot {0.5}^{n-1};[\/latex] First 4 terms: [latex]\\begin{array}{cc}400,200,100,50;& {a}_{8}=3.125\\end{array}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135187388\">\n<div id=\"fs-id1165135187390\">\n<p id=\"fs-id1165135187392\">Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137740810\">\n<dt>common ratio<\/dt>\n<dd id=\"fs-id1165137849293\">the ratio between any two consecutive terms in a geometric sequence<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137611024\">\n<dt>geometric sequence<\/dt>\n<dd id=\"fs-id1165137673421\">a sequence in which the ratio of a term to a previous term is a constant<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":33,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1363","chapter","type-chapter","status-publish","hentry"],"part":1359,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/trigonometry\/wp-json\/pressbooks\/v2\/chapters\/1363","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/trigonometry\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/trigonometry\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/trigonometry\/wp-json\/wp\/v2\/users\/33"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.ccconline.org\/trigonometry\/wp-json\/pressbooks\/v2\/chapters\/1363\/revisions"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/trigonometry\/wp-json\/pressbooks\/v2\/parts\/1359"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/trigonometry\/wp-json\/pressbooks\/v2\/chapters\/1363\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/trigonometry\/wp-json\/wp\/v2\/media?parent=1363"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/trigonometry\/wp-json\/pressbooks\/v2\/chapter-type?post=1363"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/trigonometry\/wp-json\/wp\/v2\/contributor?post=1363"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/trigonometry\/wp-json\/wp\/v2\/license?post=1363"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}