{"id":2708,"date":"2024-08-12T22:51:28","date_gmt":"2024-08-12T22:51:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2708"},"modified":"2024-11-25T20:12:52","modified_gmt":"2024-11-25T20:12:52","slug":"arithmetic-sequences-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/arithmetic-sequences-apply-it-1\/","title":{"raw":"Arithmetic Sequences: Apply It 1","rendered":"Arithmetic Sequences: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the regular interval between terms in a simple sequence and use it to write the sequence's terms<\/li>\r\n \t<li>Use recursive and explicit formulas to represent and analyze arithmetic sequences<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solving Application Problems with Arithmetic Sequences<\/h2>\r\nIn many application problems, it often makes sense to use an initial term of [latex]{a}_{0}[\/latex] instead of [latex]{a}_{1}[\/latex]. In these problems we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:\r\n\r\n<center>[latex]{a}_{n}={a}_{0}+dn[\/latex]<\/center><section class=\"textbox example\" aria-label=\"Example\">A five-year old child receives an allowance of [latex]$1[\/latex] each week. His parents promise him an annual increase of [latex]$2 [\/latex] per week.\r\n<ol>\r\n \t<li>Write a formula for the child\u2019s weekly allowance in a given year.<\/li>\r\n \t<li>What will the child\u2019s allowance be when he is [latex]16[\/latex] years old?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"752686\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"752686\"]\r\n<ol>\r\n \t<li>The situation can be modeled by an arithmetic sequence with an initial term of [latex]1[\/latex] and a common difference of [latex]2[\/latex]. Let [latex]A[\/latex] be the amount of the allowance and [latex]n[\/latex] be the number of years after age [latex]5[\/latex]. Using the altered explicit formula for an arithmetic sequence we get:\r\n<center>[latex]{A}_{n}=1+2n[\/latex]<\/center><\/li>\r\n \t<li>We can find the number of years since age [latex]5[\/latex] by subtracting.\r\n<center>[latex]16 - 5=11[\/latex]<\/center>\r\nWe are looking for the child\u2019s allowance after [latex]11[\/latex] years. Substitute [latex]11[\/latex] into the formula to find the child\u2019s allowance at age [latex]16[\/latex].\r\n<center>[latex]{A}_{11}=1+2\\left(11\\right)=23[\/latex]<\/center>\r\nThe child\u2019s allowance at age [latex]16[\/latex] will be [latex]$23[\/latex] per week.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A woman decides to go for a [latex]10[\/latex]-minute run every day this week and plans to increase the time of her daily run by [latex]4[\/latex] minutes each week. Write a formula for the time of her run after [latex]n[\/latex] weeks. How long will her daily run be [latex]8[\/latex] weeks from today?[reveal-answer q=\"356014\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"356014\"]The formula is [latex]{T}_{n}=10+4n[\/latex], and it will take her [latex]42[\/latex] minutes.[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24937[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the regular interval between terms in a simple sequence and use it to write the sequence&#8217;s terms<\/li>\n<li>Use recursive and explicit formulas to represent and analyze arithmetic sequences<\/li>\n<\/ul>\n<\/section>\n<h2>Solving Application Problems with Arithmetic Sequences<\/h2>\n<p>In many application problems, it often makes sense to use an initial term of [latex]{a}_{0}[\/latex] instead of [latex]{a}_{1}[\/latex]. In these problems we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:<\/p>\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{0}+dn[\/latex]<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A five-year old child receives an allowance of [latex]$1[\/latex] each week. His parents promise him an annual increase of [latex]$2[\/latex] per week.<\/p>\n<ol>\n<li>Write a formula for the child\u2019s weekly allowance in a given year.<\/li>\n<li>What will the child\u2019s allowance be when he is [latex]16[\/latex] years old?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q752686\">Show Solution<\/button><\/p>\n<div id=\"q752686\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The situation can be modeled by an arithmetic sequence with an initial term of [latex]1[\/latex] and a common difference of [latex]2[\/latex]. Let [latex]A[\/latex] be the amount of the allowance and [latex]n[\/latex] be the number of years after age [latex]5[\/latex]. Using the altered explicit formula for an arithmetic sequence we get:\n<div style=\"text-align: center;\">[latex]{A}_{n}=1+2n[\/latex]<\/div>\n<\/li>\n<li>We can find the number of years since age [latex]5[\/latex] by subtracting.\n<div style=\"text-align: center;\">[latex]16 - 5=11[\/latex]<\/div>\n<p>We are looking for the child\u2019s allowance after [latex]11[\/latex] years. Substitute [latex]11[\/latex] into the formula to find the child\u2019s allowance at age [latex]16[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]{A}_{11}=1+2\\left(11\\right)=23[\/latex]<\/div>\n<p>The child\u2019s allowance at age [latex]16[\/latex] will be [latex]$23[\/latex] per week.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A woman decides to go for a [latex]10[\/latex]-minute run every day this week and plans to increase the time of her daily run by [latex]4[\/latex] minutes each week. Write a formula for the time of her run after [latex]n[\/latex] weeks. How long will her daily run be [latex]8[\/latex] weeks from today?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q356014\">Show Solution<\/button><\/p>\n<div id=\"q356014\" class=\"hidden-answer\" style=\"display: none\">The formula is [latex]{T}_{n}=10+4n[\/latex], and it will take her [latex]42[\/latex] minutes.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24937\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24937&theme=lumen&iframe_resize_id=ohm24937&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":14,"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\/2708"}],"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":7,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2708\/revisions"}],"predecessor-version":[{"id":6484,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2708\/revisions\/6484"}],"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\/2708\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2708"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2708"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2708"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2708"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}