{"id":159,"date":"2025-02-13T22:44:27","date_gmt":"2025-02-13T22:44:27","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/geometric-sequences-2\/"},"modified":"2026-03-25T21:57:37","modified_gmt":"2026-03-25T21:57:37","slug":"geometric-sequences-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/geometric-sequences-2\/","title":{"raw":"Geometric Sequences: Learn It 1","rendered":"Geometric Sequences: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Find the common ratio for a geometric sequence.<\/li>\r\n \t<li style=\"font-weight: 400;\">Write the formula for a geometric sequence.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use geometric sequences to solve realistic scenarios<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Geometric Sequence<\/h2>\r\n<p class=\"whitespace-pre-wrap break-words\">Many companies offer an annual cost-of-living increase to keep salaries consistent with inflation. Let's consider, for example, a recent college graduate who lands a position as a junior software developer with a starting annual salary of $[latex]54,000[\/latex]. The company promises a [latex]2 \\%[\/latex] cost of living increase each year.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">To calculate the developer's salary in any given year, we multiply their salary from the previous year by [latex]102 \\%[\/latex]. Let's see how this plays out over the first few years:<\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">After one year: [latex]$54,000 \\times 1.02 = $55,080[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">After two years: [latex]$55,080 \\times 1.02 = $56,181.60[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">After three years: [latex]$56,181.60 \\times 1.02 = $57,305.23[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\">As we can see, when a salary increases by a constant rate each year, it grows by a constant factor. In this case, the factor is [latex]1.02[\/latex] or [latex]102 \\%[\/latex] of the previous year's salary.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">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>.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>geometric sequence<\/h3>\r\nA sequence is called a <strong>geometric sequence<\/strong> if the ratio between consecutive terms is always the same.\r\nThe ratio between consecutive terms in a geometric sequence is [latex]r[\/latex], the <strong>common ratio<\/strong>, where [latex]n \\ge 2[\/latex].\r\n<p style=\"text-align: center;\">[latex]r = \\dfrac{a_n}{a_{n-1}}[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThe geometric sequence will be\r\n<p style=\"text-align: center;\">[latex]\\left\\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\\right\\}[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">The sequence below is an example of a geometric sequence because each term increases by a constant factor of [latex]6[\/latex]. Multiplying any term of the sequence by the common ratio [latex]r = 6[\/latex] generates the subsequent term.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03223627\/CNX_Precalc_Figure_11_03_0012.jpg\" alt=\"A sequence , {1, 6, 36, 216, 1296, ...} that shows all the numbers have a common ratio of 6.\" \/><\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a set of numbers, determine if they represent a geometric sequence.<\/strong>\r\n<ol>\r\n \t<li>Divide each term by the previous term.<\/li>\r\n \t<li>Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Determine if the follow sequences are geometric? If so, find the common ratio.\r\n<ul>\r\n \t<li>[latex]4, 8, 16, 32, 64, 128, \\dots[\/latex]\r\n[reveal-answer q=\"687536\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"687536\"]<img class=\"aligncenter wp-image-2721\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/12232350\/Screenshot-2024-08-12-at-4.23.46%E2%80%AFPM.png\" alt=\"\" width=\"548\" height=\"126\" \/>The sequence is geometric because there is a common ratio. The common ratio is [latex]2[\/latex].\r\n[\/hidden-answer]<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li>[latex]-2, 6, -12, 36, -72, 216, \\dots[\/latex]\r\n[reveal-answer q=\"348568\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"348568\"]\r\n<img class=\"aligncenter wp-image-2722\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/12232522\/Screenshot-2024-08-12-at-4.25.18%E2%80%AFPM.png\" alt=\"\" width=\"613\" height=\"131\" \/>The sequence is not geometric. There is no common ratio. [\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321965[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]321966[\/ohm_question]<\/section><\/div>\r\n<dl id=\"fs-id1165137611024\" class=\"definition\">\r\n \t<dd id=\"fs-id1165137673421\"><\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Find the common ratio for a geometric sequence.<\/li>\n<li style=\"font-weight: 400;\">Write the formula for a geometric sequence.<\/li>\n<li style=\"font-weight: 400;\">Use geometric sequences to solve realistic scenarios<\/li>\n<\/ul>\n<\/section>\n<h2>Geometric Sequence<\/h2>\n<p class=\"whitespace-pre-wrap break-words\">Many companies offer an annual cost-of-living increase to keep salaries consistent with inflation. Let&#8217;s consider, for example, a recent college graduate who lands a position as a junior software developer with a starting annual salary of $[latex]54,000[\/latex]. The company promises a [latex]2 \\%[\/latex] cost of living increase each year.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">To calculate the developer&#8217;s salary in any given year, we multiply their salary from the previous year by [latex]102 \\%[\/latex]. Let&#8217;s see how this plays out over the first few years:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">After one year: [latex]$54,000 \\times 1.02 = $55,080[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">After two years: [latex]$55,080 \\times 1.02 = $56,181.60[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">After three years: [latex]$56,181.60 \\times 1.02 = $57,305.23[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\">As we can see, when a salary increases by a constant rate each year, it grows by a constant factor. In this case, the factor is [latex]1.02[\/latex] or [latex]102 \\%[\/latex] of the previous year&#8217;s salary.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">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>.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>geometric sequence<\/h3>\n<p>A sequence is called a <strong>geometric sequence<\/strong> if the ratio between consecutive terms is always the same.<br \/>\nThe ratio between consecutive terms in a geometric sequence is [latex]r[\/latex], the <strong>common ratio<\/strong>, where [latex]n \\ge 2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]r = \\dfrac{a_n}{a_{n-1}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The geometric sequence will be<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\\right\\}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">The sequence below is an example of a geometric sequence because each term increases by a constant factor of [latex]6[\/latex]. Multiplying any term of the sequence by the common ratio [latex]r = 6[\/latex] generates the subsequent term.<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03223627\/CNX_Precalc_Figure_11_03_0012.jpg\" alt=\"A sequence , {1, 6, 36, 216, 1296, ...} that shows all the numbers have a common ratio of 6.\" \/><\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a set of numbers, determine if they represent a geometric sequence.<\/strong><\/p>\n<ol>\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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Determine if the follow sequences are geometric? If so, find the common ratio.<\/p>\n<ul>\n<li>[latex]4, 8, 16, 32, 64, 128, \\dots[\/latex]\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q687536\">Show Answer<\/button><\/p>\n<div id=\"q687536\" class=\"hidden-answer\" style=\"display: none\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2721\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/12232350\/Screenshot-2024-08-12-at-4.23.46%E2%80%AFPM.png\" alt=\"\" width=\"548\" height=\"126\" \/>The sequence is geometric because there is a common ratio. The common ratio is [latex]2[\/latex].\n<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<ul>\n<li>[latex]-2, 6, -12, 36, -72, 216, \\dots[\/latex]\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q348568\">Show Answer<\/button><\/p>\n<div id=\"q348568\" class=\"hidden-answer\" style=\"display: none\">\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2722\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/12232522\/Screenshot-2024-08-12-at-4.25.18%E2%80%AFPM.png\" alt=\"\" width=\"613\" height=\"131\" \/>The sequence is not geometric. There is no common ratio. <\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321965\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321965&theme=lumen&iframe_resize_id=ohm321965&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm321966\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321966&theme=lumen&iframe_resize_id=ohm321966&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<dl id=\"fs-id1165137611024\" class=\"definition\">\n<dd id=\"fs-id1165137673421\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":156,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/159"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/159\/revisions"}],"predecessor-version":[{"id":6029,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/159\/revisions\/6029"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/156"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/159\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=159"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=159"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=159"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=159"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}