{"id":2735,"date":"2024-08-13T00:20:17","date_gmt":"2024-08-13T00:20:17","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2735"},"modified":"2024-12-02T19:02:47","modified_gmt":"2024-12-02T19:02:47","slug":"geometric-sequences-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/geometric-sequences-apply-it-1\/","title":{"raw":"Geometric Sequences: Apply It 1","rendered":"Geometric Sequences: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Determine if a sequence is geometric, find the common ratio, list the terms, and find the general (nth) term of a geometric sequence<\/li>\r\n \t<li>Use recursive and explicit formulas to describe and study geometric sequences<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solving Application Problems with Geometric Sequences<\/h2>\r\nIn 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:\r\n<p style=\"text-align: center;\">[latex]{a}_{n}={a}_{0}{r}^{n}[\/latex]<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">In 2021, the number of students in a small school is [latex]284[\/latex]. It is estimated that the student population will increase by [latex]4 \\%[\/latex] each year.\r\n<ol>\r\n \t<li>Write a formula for the student population.<\/li>\r\n \t<li>Estimate the student population in 2028.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"304368\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"304368\"]\r\n<ol>\r\n \t<li>The situation can be modeled by a geometric sequence with an initial term of [latex]284[\/latex]. The student population will be [latex]104 \\%[\/latex] of the prior year, so the common ratio is [latex]1.04[\/latex].Let [latex]P[\/latex] be the student population and [latex]n[\/latex] be the number of years after 2021. Using the explicit formula for a geometric sequence we get<center>[latex]{P}_{n} =284\\cdot {1.04}^{n}[\/latex]<\/center><\/li>\r\n \t<li>We can find the number of years since 2021 by subtracting.\r\n<center>[latex]2028 - 2021=7[\/latex]<\/center>\r\nWe are looking for the population after [latex]7[\/latex] years. We can substitute [latex]7[\/latex] for [latex]n[\/latex] to estimate the population in 2028.\r\n<center>[latex]{P}_{7}=284\\cdot {1.04}^{7}\\approx 374[\/latex]<\/center>\r\nThe student population will be about [latex]374[\/latex] in 2028.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A new investor places [latex]$5,000[\/latex] in a high-yield savings account that offers [latex]6 \\%[\/latex] annual interest, compounded annually. Assuming she doesn't make any additional deposits or withdrawals, how much money will be in the account after [latex]10[\/latex] years?[reveal-answer q=\"297652\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"297652\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the geometric sequence:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Initial amount [latex]a_1 = $5,000[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Common ratio [latex]r = 1 + 0.06 = 1.06[\/latex] ([latex]106 \\%[\/latex] of previous year's amount)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">We want to find the [latex]10[\/latex]th term [latex](n = 10)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use the geometric sequence formula:\r\n<center>[latex]a_n = a_1 \\cdot r^{n-1}[\/latex]<\/center>\r\n<center>[latex]a_{10} = 5000 \\cdot (1.06)^{10-1}[\/latex]<\/center><\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate:\r\n<center>[latex]\r\n\\begin{array}{rcl}\r\na_{10} &amp;=&amp; 5000 \\cdot (1.06)^9 \\\\\r\n&amp;=&amp; 5000 \\cdot 1.6895 \\\\\r\n&amp;\\approx&amp; $8,447.50\r\n\\end{array}\r\n[\/latex]<\/center><\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, after [latex]10[\/latex] years, the account will contain approximately [latex]$8,447.50[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A new car is purchased for [latex]$32,000[\/latex]. Each year, its value decreases by [latex]15 \\%[\/latex] of its value from the previous year. What will be the car's value after [latex]5[\/latex] years?[reveal-answer q=\"34154\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"34154\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the geometric sequence:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Initial value [latex]a_1 = $32,000[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Common ratio [latex]r = 1 - 0.15 = 0.85[\/latex] ([latex]85 \\%[\/latex] of previous year's value)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">We want to find the [latex]5[\/latex]th term [latex](n = 5)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use the geometric sequence formula:\r\n<center>[latex]a_n = a_1 \\cdot r^{n-1}[\/latex]<\/center>\r\n<center>[latex]a_5 = 32000 \\cdot (0.85)^{5-1}[\/latex]<\/center><\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate:\r\n<center>[latex]\r\n\\begin{array}{rcl}\r\na_5 &amp;=&amp; 32000 \\cdot (0.85)^4 \\\\\r\n&amp;=&amp; 32000 \\cdot 0.522 \\\\\r\n&amp;\\approx&amp; $16,704\r\n\\end{array}\r\n[\/latex]<\/center><\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, after [latex]5[\/latex] years, the car's value will be approximately [latex]$16,704[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]291406[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Determine if a sequence is geometric, find the common ratio, list the terms, and find the general (nth) term of a geometric sequence<\/li>\n<li>Use recursive and explicit formulas to describe and study geometric sequences<\/li>\n<\/ul>\n<\/section>\n<h2>Solving Application Problems with Geometric Sequences<\/h2>\n<p>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<p style=\"text-align: center;\">[latex]{a}_{n}={a}_{0}{r}^{n}[\/latex]<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">In 2021, the number of students in a small school is [latex]284[\/latex]. It is estimated that the student population will increase by [latex]4 \\%[\/latex] each year.<\/p>\n<ol>\n<li>Write a formula for the student population.<\/li>\n<li>Estimate the student population in 2028.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q304368\">Show Solution<\/button><\/p>\n<div id=\"q304368\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The situation can be modeled by a geometric sequence with an initial term of [latex]284[\/latex]. The student population will be [latex]104 \\%[\/latex] of the prior year, so the common ratio is [latex]1.04[\/latex].Let [latex]P[\/latex] be the student population and [latex]n[\/latex] be the number of years after 2021. Using the explicit formula for a geometric sequence we get\n<div style=\"text-align: center;\">[latex]{P}_{n} =284\\cdot {1.04}^{n}[\/latex]<\/div>\n<\/li>\n<li>We can find the number of years since 2021 by subtracting.\n<div style=\"text-align: center;\">[latex]2028 - 2021=7[\/latex]<\/div>\n<p>We are looking for the population after [latex]7[\/latex] years. We can substitute [latex]7[\/latex] for [latex]n[\/latex] to estimate the population in 2028.<\/p>\n<div style=\"text-align: center;\">[latex]{P}_{7}=284\\cdot {1.04}^{7}\\approx 374[\/latex]<\/div>\n<p>The student population will be about [latex]374[\/latex] in 2028.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A new investor places [latex]$5,000[\/latex] in a high-yield savings account that offers [latex]6 \\%[\/latex] annual interest, compounded annually. Assuming she doesn&#8217;t make any additional deposits or withdrawals, how much money will be in the account after [latex]10[\/latex] years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q297652\">Show Answer<\/button><\/p>\n<div id=\"q297652\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the geometric sequence:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Initial amount [latex]a_1 = $5,000[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Common ratio [latex]r = 1 + 0.06 = 1.06[\/latex] ([latex]106 \\%[\/latex] of previous year&#8217;s amount)<\/li>\n<li class=\"whitespace-normal break-words\">We want to find the [latex]10[\/latex]th term [latex](n = 10)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Use the geometric sequence formula:\n<div style=\"text-align: center;\">[latex]a_n = a_1 \\cdot r^{n-1}[\/latex]<\/div>\n<div style=\"text-align: center;\">[latex]a_{10} = 5000 \\cdot (1.06)^{10-1}[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Calculate:\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl}  a_{10} &=& 5000 \\cdot (1.06)^9 \\\\  &=& 5000 \\cdot 1.6895 \\\\  &\\approx& $8,447.50  \\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, after [latex]10[\/latex] years, the account will contain approximately [latex]$8,447.50[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A new car is purchased for [latex]$32,000[\/latex]. Each year, its value decreases by [latex]15 \\%[\/latex] of its value from the previous year. What will be the car&#8217;s value after [latex]5[\/latex] years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q34154\">Show Answer<\/button><\/p>\n<div id=\"q34154\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the geometric sequence:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Initial value [latex]a_1 = $32,000[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Common ratio [latex]r = 1 - 0.15 = 0.85[\/latex] ([latex]85 \\%[\/latex] of previous year&#8217;s value)<\/li>\n<li class=\"whitespace-normal break-words\">We want to find the [latex]5[\/latex]th term [latex](n = 5)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Use the geometric sequence formula:\n<div style=\"text-align: center;\">[latex]a_n = a_1 \\cdot r^{n-1}[\/latex]<\/div>\n<div style=\"text-align: center;\">[latex]a_5 = 32000 \\cdot (0.85)^{5-1}[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Calculate:\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl}  a_5 &=& 32000 \\cdot (0.85)^4 \\\\  &=& 32000 \\cdot 0.522 \\\\  &\\approx& $16,704  \\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, after [latex]5[\/latex] years, the car&#8217;s value will be approximately [latex]$16,704[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm291406\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=291406&theme=lumen&iframe_resize_id=ohm291406&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":20,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":363,"module-header":"apply_it","content_attributions":[],"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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2735"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":15,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2735\/revisions"}],"predecessor-version":[{"id":6536,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2735\/revisions\/6536"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/363"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2735\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2735"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2735"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2735"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2735"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}