{"id":1487,"date":"2025-07-25T02:10:32","date_gmt":"2025-07-25T02:10:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1487"},"modified":"2026-03-24T07:12:06","modified_gmt":"2026-03-24T07:12:06","slug":"geometric-sequences-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/geometric-sequences-fresh-take\/","title":{"raw":"Geometric Sequences: Fresh Take","rendered":"Geometric Sequences: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the common ratio for a geometric sequence.<\/li>\r\n \t<li>Write the formula for a geometric sequence.<\/li>\r\n \t<li>Use geometric sequences to solve realistic scenarios<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Geometric Sequence<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition: A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Form: [latex]{a_n} = {a_1, a_1r, a_1r^2, a_1r^3, ..., a_1r^{n-1}}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Common Ratio: [latex]r = \\frac{a_n}{a_{n-1}}[\/latex] for any [latex]n \\geq 2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Explicit Formula: [latex]a_n = a_1r^{n-1}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Properties:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Can model exponential growth or decay<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Terms can be positive, negative, or alternating<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The absolute values of terms either strictly increase, strictly decrease, or remain constant<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Real-world Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Compound interest<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Population growth<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Radioactive decay<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Moore's Law in computing<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Determine if the follow sequences are geometric? If so, find the common ratio.\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>[latex]5,10,15,20,\\dots[\/latex]<\/li>\r\n \t<li>[latex]48,12,4,2,\\dots[\/latex]<\/li>\r\n \t<li>[latex]100,20,4,\\dfrac{4}{5},\\dots[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"893960\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"893960\"]\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>The sequence is not geometric because [latex]\\dfrac{10}{5}\\ne \\dfrac{15}{10}[\/latex] .<\/li>\r\n \t<li>[latex]\\begin{align}&amp;\\frac{12}{48}=\\frac{1}{4} &amp;&amp; \\frac{4}{12}=\\frac{1}{3} &amp;&amp; \\frac{2}{4}=\\frac{1}{2} \\end{align}[\/latex]\r\nThe sequence is not geometric because there is not a common ratio.<\/li>\r\n \t<li>The sequence is geometric. The common ratio is [latex]\\dfrac{1}{5}[\/latex] .<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Writing Terms of Geometric Sequences<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Term Generation Formula: [latex]a_n = a_1 \\cdot r^{n-1}[\/latex] Where:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_n[\/latex] is the [latex]n[\/latex]th term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex] is the first term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex] is the common ratio<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is the term number<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Recursive Process: Each term is the product of the previous term and the common ratio: [latex]a_n = a_{n-1} \\cdot r[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Pattern Recognition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If [latex]|r| &gt; 1[\/latex], the absolute values of terms increase<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]0 &lt; |r| &lt; 1[\/latex], the absolute values of terms decrease<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]r &lt; 0[\/latex], the terms alternate in sign<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Exponential Nature: The exponent of r in each term is one less than the term's position<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\dfrac{1}{3}[\/latex].[reveal-answer q=\"228021\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"228021\"][latex]\\left\\{18,6,2,\\dfrac{2}{3},\\dfrac{2}{9}\\right\\}[\/latex][\/hidden-answer]<\/section>\r\n<h2>Using Explicit Formulas for Geometric Sequences<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Geometric Sequence as an Exponential Function:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A geometric sequence is an exponential function with a domain of positive integers.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The common ratio (r) is the base of the function.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Explicit Formula:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex]a_n = a_1r^{n-1}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_n[\/latex]: nth term of the sequence<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex]: first term of the sequence<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: common ratio<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]n[\/latex]: term number<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].[reveal-answer q=\"132711\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"132711\"][latex]{a}_{6}=16\\text{,}384[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Write an explicit formula for the following geometric sequence.\r\n<p style=\"text-align: center;\">[latex]\\left\\{-1,3,-9,27,\\dots\\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"521311\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"521311\"]\r\n\r\n[latex]{a}_{n}=-{\\left(-3\\right)}^{n - 1}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Using Recursive Formulas for Geometric Sequences<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Recursive Formula Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Allows finding any term using the previous term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each term is the product of the common ratio and the previous term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Initial term must be given<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Recursive Formula:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_n = ra_{n-1}, n \\ge 2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: common ratio<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_n[\/latex]: nth term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_{n-1}[\/latex]: previous term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex]: first term (must be specified)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write a recursive formula for the following geometric sequence.[latex]\\left\\{2,\\dfrac{4}{3},\\dfrac{8}{9},\\dfrac{16}{27},\\dots\\right\\}[\/latex][reveal-answer q=\"483012\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"483012\"][latex]\\begin{align}&amp;{a}_{1}=2\\\\ &amp;{a}_{n}=\\frac{2}{3}\\cdot{a}_{n - 1}\\text{ for }n\\ge 2\\end{align}[\/latex][\/hidden-answer]<\/section>Watch the following videos for more examples involving geometric sequences.\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bddgcaae-XHyeLKZYb2w\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/XHyeLKZYb2w?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bddgcaae-XHyeLKZYb2w\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851336&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bddgcaae-XHyeLKZYb2w&amp;vembed=0&amp;video_id=XHyeLKZYb2w&amp;video_target=tpm-plugin-bddgcaae-XHyeLKZYb2w\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Geometric+Sequences_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGeometric Sequences\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cdhgfchc-LbQ3CW9AofU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/LbQ3CW9AofU?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cdhgfchc-LbQ3CW9AofU\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851337&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cdhgfchc-LbQ3CW9AofU&amp;vembed=0&amp;video_id=LbQ3CW9AofU&amp;video_target=tpm-plugin-cdhgfchc-LbQ3CW9AofU\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Determine+the+Type+of+Sequence+Given+a+Sequence+Formula_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Determine the Type of Sequence Given a Sequence Formula\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-efhgbgge-S8qsbzZiRqQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/S8qsbzZiRqQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-efhgbgge-S8qsbzZiRqQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851338&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-efhgbgge-S8qsbzZiRqQ&amp;vembed=0&amp;video_id=S8qsbzZiRqQ&amp;video_target=tpm-plugin-efhgbgge-S8qsbzZiRqQ\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Find+the+Formula+for+a+Sequence+Given+Terms+(Arithmetic+and+Geometric)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Find the Formula for a Sequence Given Terms (Arithmetic and Geometric)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the common ratio for a geometric sequence.<\/li>\n<li>Write the formula for a geometric sequence.<\/li>\n<li>Use geometric sequences to solve realistic scenarios<\/li>\n<\/ul>\n<\/section>\n<h2>Geometric Sequence<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition: A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.<\/li>\n<li class=\"whitespace-normal break-words\">General Form: [latex]{a_n} = {a_1, a_1r, a_1r^2, a_1r^3, ..., a_1r^{n-1}}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Common Ratio: [latex]r = \\frac{a_n}{a_{n-1}}[\/latex] for any [latex]n \\geq 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Explicit Formula: [latex]a_n = a_1r^{n-1}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Key Properties:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Can model exponential growth or decay<\/li>\n<li class=\"whitespace-normal break-words\">Terms can be positive, negative, or alternating<\/li>\n<li class=\"whitespace-normal break-words\">The absolute values of terms either strictly increase, strictly decrease, or remain constant<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Real-world Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Compound interest<\/li>\n<li class=\"whitespace-normal break-words\">Population growth<\/li>\n<li class=\"whitespace-normal break-words\">Radioactive decay<\/li>\n<li class=\"whitespace-normal break-words\">Moore&#8217;s Law in computing<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Determine if the follow sequences are geometric? If so, find the common ratio.<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]5,10,15,20,\\dots[\/latex]<\/li>\n<li>[latex]48,12,4,2,\\dots[\/latex]<\/li>\n<li>[latex]100,20,4,\\dfrac{4}{5},\\dots[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q893960\">Show Solution<\/button><\/p>\n<div id=\"q893960\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: decimal;\">\n<li>The sequence is not geometric because [latex]\\dfrac{10}{5}\\ne \\dfrac{15}{10}[\/latex] .<\/li>\n<li>[latex]\\begin{align}&\\frac{12}{48}=\\frac{1}{4} && \\frac{4}{12}=\\frac{1}{3} && \\frac{2}{4}=\\frac{1}{2} \\end{align}[\/latex]<br \/>\nThe sequence is not geometric because there is not a common ratio.<\/li>\n<li>The sequence is geometric. The common ratio is [latex]\\dfrac{1}{5}[\/latex] .<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>Writing Terms of Geometric Sequences<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Term Generation Formula: [latex]a_n = a_1 \\cdot r^{n-1}[\/latex] Where:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a_n[\/latex] is the [latex]n[\/latex]th term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex] is the first term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex] is the common ratio<\/li>\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is the term number<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Recursive Process: Each term is the product of the previous term and the common ratio: [latex]a_n = a_{n-1} \\cdot r[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Pattern Recognition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]|r| > 1[\/latex], the absolute values of terms increase<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]0 < |r| < 1[\/latex], the absolute values of terms decrease<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]r < 0[\/latex], the terms alternate in sign<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Exponential Nature: The exponent of r in each term is one less than the term&#8217;s position<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">List the first five terms of the geometric sequence with [latex]{a}_{1}=18[\/latex] and [latex]r=\\dfrac{1}{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q228021\">Show Solution<\/button><\/p>\n<div id=\"q228021\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left\\{18,6,2,\\dfrac{2}{3},\\dfrac{2}{9}\\right\\}[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Using Explicit Formulas for Geometric Sequences<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Geometric Sequence as an Exponential Function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A geometric sequence is an exponential function with a domain of positive integers.<\/li>\n<li class=\"whitespace-normal break-words\">The common ratio (r) is the base of the function.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Explicit Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">General form: [latex]a_n = a_1r^{n-1}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_n[\/latex]: nth term of the sequence<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex]: first term of the sequence<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: common ratio<\/li>\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex]: term number<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Given a geometric sequence with [latex]{a}_{2}=4[\/latex] and [latex]{a}_{3}=32[\/latex] , find [latex]{a}_{6}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q132711\">Show Solution<\/button><\/p>\n<div id=\"q132711\" class=\"hidden-answer\" style=\"display: none\">[latex]{a}_{6}=16\\text{,}384[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write an explicit formula for the following geometric sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{-1,3,-9,27,\\dots\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q521311\">Show Solution<\/button><\/p>\n<div id=\"q521311\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{n}=-{\\left(-3\\right)}^{n - 1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Using Recursive Formulas for Geometric Sequences<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Recursive Formula Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Allows finding any term using the previous term<\/li>\n<li class=\"whitespace-normal break-words\">Each term is the product of the common ratio and the previous term<\/li>\n<li class=\"whitespace-normal break-words\">Initial term must be given<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Recursive Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a_n = ra_{n-1}, n \\ge 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex]: common ratio<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_n[\/latex]: nth term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_{n-1}[\/latex]: previous term<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a_1[\/latex]: first term (must be specified)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write a recursive formula for the following geometric sequence.[latex]\\left\\{2,\\dfrac{4}{3},\\dfrac{8}{9},\\dfrac{16}{27},\\dots\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q483012\">Show Solution<\/button><\/p>\n<div id=\"q483012\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{align}&{a}_{1}=2\\\\ &{a}_{n}=\\frac{2}{3}\\cdot{a}_{n - 1}\\text{ for }n\\ge 2\\end{align}[\/latex]<\/div>\n<\/div>\n<\/section>\n<p>Watch the following videos for more examples involving geometric sequences.<\/p>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bddgcaae-XHyeLKZYb2w\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/XHyeLKZYb2w?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bddgcaae-XHyeLKZYb2w\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851336&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-bddgcaae-XHyeLKZYb2w&amp;vembed=0&amp;video_id=XHyeLKZYb2w&amp;video_target=tpm-plugin-bddgcaae-XHyeLKZYb2w\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Geometric+Sequences_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGeometric Sequences\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cdhgfchc-LbQ3CW9AofU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/LbQ3CW9AofU?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cdhgfchc-LbQ3CW9AofU\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851337&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cdhgfchc-LbQ3CW9AofU&amp;vembed=0&amp;video_id=LbQ3CW9AofU&amp;video_target=tpm-plugin-cdhgfchc-LbQ3CW9AofU\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Determine+the+Type+of+Sequence+Given+a+Sequence+Formula_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Determine the Type of Sequence Given a Sequence Formula\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-efhgbgge-S8qsbzZiRqQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/S8qsbzZiRqQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-efhgbgge-S8qsbzZiRqQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851338&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-efhgbgge-S8qsbzZiRqQ&amp;vembed=0&amp;video_id=S8qsbzZiRqQ&amp;video_target=tpm-plugin-efhgbgge-S8qsbzZiRqQ\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Find+the+Formula+for+a+Sequence+Given+Terms+(Arithmetic+and+Geometric)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Find the Formula for a Sequence Given Terms (Arithmetic and Geometric)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Geometric Sequences\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/XHyeLKZYb2w\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Determine the Type of Sequence Given a Sequence Formula\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/LbQ3CW9AofU\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 2: Find the Formula for a Sequence Given Terms (Arithmetic and Geometric)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/S8qsbzZiRqQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube 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