{"id":1742,"date":"2025-07-28T17:56:56","date_gmt":"2025-07-28T17:56:56","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1742"},"modified":"2026-03-26T16:36:14","modified_gmt":"2026-03-26T16:36:14","slug":"series-and-their-notations-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/series-and-their-notations-apply-it\/","title":{"raw":"Series and Their Notations: Apply It","rendered":"Series and Their Notations: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use summation notation to represent a series.<\/li>\r\n \t<li>Use the formula for the sum of the \ufb01rst n terms of an arithmetic series.<\/li>\r\n \t<li>Use the formula for the sum of the \ufb01rst n terms of a geometric series.<\/li>\r\n \t<li>Use the formula for the sum of an in\ufb01nite geometric series.<\/li>\r\n \t<li>Solve word problems involving series.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solving Annuity Problems<\/h2>\r\nConsider a couple who invested a set amount of money each month into a college fund for six years.\r\n\r\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments.<\/section>To find the amount of an annuity, we need to find the sum of all the payments and the interest earned.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Suppose the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.We can find the value of the annuity right after the last deposit by using a geometric series with [latex]{a}_{1}=50[\/latex] and [latex]r=100.5%=1.005[\/latex]. After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.We can find the value of the annuity after [latex]n[\/latex] deposits using the formula for the sum of the first [latex]n[\/latex] terms of a geometric series. In 6 years, there are 72 months, so [latex]n=72[\/latex]. We can substitute [latex]{a}_{1}=50, r=1.005, \\text{and} n=72[\/latex] into the formula, and simplify to find the value of the annuity after 6 years.\r\n<div style=\"text-align: center;\">[latex]{S}_{72}=\\frac{50\\left(1-{1.005}^{72}\\right)}{1 - 1.005}\\approx 4\\text{,}320.44[\/latex]<\/div>\r\nAfter the last deposit, the couple will have a total of $4,320.44 in the account. Notice, the couple made 72 payments of $50 each for a total of [latex]72\\left(50\\right) = $3,600[\/latex]. This means that because of the annuity, the couple earned $720.44 interest in their college fund.\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given an initial deposit and an interest rate, find the value of an annuity.<\/strong>\r\n<ol>\r\n \t<li>Determine [latex]{a}_{1}[\/latex], the value of the initial deposit.<\/li>\r\n \t<li>Determine [latex]n[\/latex], the number of deposits.<\/li>\r\n \t<li>Determine [latex]r[\/latex].\r\n<ol>\r\n \t<li>Divide the annual interest rate by the number of times per year that interest is compounded.<\/li>\r\n \t<li>Add 1 to this amount to find [latex]r[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Substitute values for [latex]{a}_{1}\\text{,}r,\\text{ and }n[\/latex]\r\ninto the formula for the sum of the first [latex]n[\/latex] terms of a geometric series, [latex]{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/latex].<\/li>\r\n \t<li>Simplify to find [latex]{S}_{n}[\/latex], the value of the annuity after [latex]n[\/latex] deposits.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A deposit of $100 is placed into a college fund at the beginning of every month for 10 years. The fund earns 9% annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit?[reveal-answer q=\"385976\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"385976\"]The value of the initial deposit is $100, so [latex]{a}_{1}=100[\/latex]. A total of 120 monthly deposits are made in the 10 years, so [latex]n=120[\/latex]. To find [latex]r[\/latex], divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new monthly deposit.\r\n<div style=\"text-align: center;\">[latex]r=1+\\frac{0.09}{12}=1.0075[\/latex]<\/div>\r\nSubstitute [latex]{a}_{1}=100\\text{,}r=1.0075\\text{,}\\text{and}n=120[\/latex] into the formula for the sum of the first [latex]n[\/latex] terms of a geometric series, and simplify to find the value of the annuity.\r\n<div style=\"text-align: center;\">[latex]{S}_{120}=\\frac{100\\left(1-{1.0075}^{120}\\right)}{1 - 1.0075}\\approx 19\\text{,}351.43[\/latex]<\/div>\r\nSo the account has $19,351.43 after the last deposit is made.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div class=\"bcc-box bcc-success\">\r\n\r\n[ohm_question hide_question_numbers=1]321987[\/ohm_question]\r\n\r\n<\/div>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use summation notation to represent a series.<\/li>\n<li>Use the formula for the sum of the \ufb01rst n terms of an arithmetic series.<\/li>\n<li>Use the formula for the sum of the \ufb01rst n terms of a geometric series.<\/li>\n<li>Use the formula for the sum of an in\ufb01nite geometric series.<\/li>\n<li>Solve word problems involving series.<\/li>\n<\/ul>\n<\/section>\n<h2>Solving Annuity Problems<\/h2>\n<p>Consider a couple who invested a set amount of money each month into a college fund for six years.<\/p>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments.<\/section>\n<p>To find the amount of an annuity, we need to find the sum of all the payments and the interest earned.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Suppose the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.We can find the value of the annuity right after the last deposit by using a geometric series with [latex]{a}_{1}=50[\/latex] and [latex]r=100.5%=1.005[\/latex]. After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.We can find the value of the annuity after [latex]n[\/latex] deposits using the formula for the sum of the first [latex]n[\/latex] terms of a geometric series. In 6 years, there are 72 months, so [latex]n=72[\/latex]. We can substitute [latex]{a}_{1}=50, r=1.005, \\text{and} n=72[\/latex] into the formula, and simplify to find the value of the annuity after 6 years.<\/p>\n<div style=\"text-align: center;\">[latex]{S}_{72}=\\frac{50\\left(1-{1.005}^{72}\\right)}{1 - 1.005}\\approx 4\\text{,}320.44[\/latex]<\/div>\n<p>After the last deposit, the couple will have a total of $4,320.44 in the account. Notice, the couple made 72 payments of $50 each for a total of [latex]72\\left(50\\right) = $3,600[\/latex]. This means that because of the annuity, the couple earned $720.44 interest in their college fund.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given an initial deposit and an interest rate, find the value of an annuity.<\/strong><\/p>\n<ol>\n<li>Determine [latex]{a}_{1}[\/latex], the value of the initial deposit.<\/li>\n<li>Determine [latex]n[\/latex], the number of deposits.<\/li>\n<li>Determine [latex]r[\/latex].\n<ol>\n<li>Divide the annual interest rate by the number of times per year that interest is compounded.<\/li>\n<li>Add 1 to this amount to find [latex]r[\/latex].<\/li>\n<\/ol>\n<\/li>\n<li>Substitute values for [latex]{a}_{1}\\text{,}r,\\text{ and }n[\/latex]<br \/>\ninto the formula for the sum of the first [latex]n[\/latex] terms of a geometric series, [latex]{S}_{n}=\\frac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/latex].<\/li>\n<li>Simplify to find [latex]{S}_{n}[\/latex], the value of the annuity after [latex]n[\/latex] deposits.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A deposit of $100 is placed into a college fund at the beginning of every month for 10 years. The fund earns 9% annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q385976\">Show Solution<\/button><\/p>\n<div id=\"q385976\" class=\"hidden-answer\" style=\"display: none\">The value of the initial deposit is $100, so [latex]{a}_{1}=100[\/latex]. A total of 120 monthly deposits are made in the 10 years, so [latex]n=120[\/latex]. To find [latex]r[\/latex], divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new monthly deposit.<\/p>\n<div style=\"text-align: center;\">[latex]r=1+\\frac{0.09}{12}=1.0075[\/latex]<\/div>\n<p>Substitute [latex]{a}_{1}=100\\text{,}r=1.0075\\text{,}\\text{and}n=120[\/latex] into the formula for the sum of the first [latex]n[\/latex] terms of a geometric series, and simplify to find the value of the annuity.<\/p>\n<div style=\"text-align: center;\">[latex]{S}_{120}=\\frac{100\\left(1-{1.0075}^{120}\\right)}{1 - 1.0075}\\approx 19\\text{,}351.43[\/latex]<\/div>\n<p>So the account has $19,351.43 after the last deposit is made.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div class=\"bcc-box bcc-success\">\n<p><iframe loading=\"lazy\" id=\"ohm321987\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=321987&theme=lumen&iframe_resize_id=ohm321987&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/section>\n","protected":false},"author":13,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":156,"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1742"}],"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\/13"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1742\/revisions"}],"predecessor-version":[{"id":6042,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1742\/revisions\/6042"}],"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\/1742\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1742"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1742"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1742"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1742"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}