{"id":8348,"date":"2023-09-29T14:40:54","date_gmt":"2023-09-29T14:40:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8348"},"modified":"2025-08-29T20:44:45","modified_gmt":"2025-08-29T20:44:45","slug":"math-in-business-background-youll-need-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/math-in-business-background-youll-need-2\/","title":{"raw":"Math in Business: Background You'll Need 2","rendered":"Math in Business: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Understand the basics of summations, including their notation, properties, and practical uses&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:5057,&quot;3&quot;:{&quot;1&quot;:0},&quot;9&quot;:0,&quot;10&quot;:0,&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Understand the basics of summations, including their notation, properties, and practical uses<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Using Summation Notation<\/h2>\r\n<p>To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a <strong>series<\/strong>. Consider, for example, the following series.<\/p>\r\n<p style=\"text-align: center;\">[latex]3+7+11+15+19+\\cdots[\/latex]<\/p>\r\n<p>The <strong>[latex]n\\text{th }[\/latex] partial sum<\/strong> of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{S}_{n}\\text{ represents the partial sum.} \\\\ &amp;{S}_{1}=3 \\\\ &amp;{S}_{2}=3+7=10 \\\\ &amp;{S}_{3}=3+7+11=21 \\\\ &amp;{S}_{4}=3+7+11+15=36\\end{align}[\/latex]<\/p>\r\n<p><strong>Summation notation <\/strong>is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter <strong>sigma<\/strong>, [latex]\\Sigma[\/latex], to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma.<\/p>\r\n<ul>\r\n\t<li>A variable called the <strong>index of summation <\/strong>is written below the sigma.<\/li>\r\n\t<li>The index of summation is set equal to the <strong>lower limit of summation<\/strong>, which is the number used to generate the first term in the series.<\/li>\r\n\t<li>The number above the sigma, called the <strong>upper limit of summation<\/strong>, is the number used to generate the last term in a series.<\/li>\r\n<\/ul>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03225157\/CNX_Precalc_Figure_11_04_001n2.jpg\" alt=\"Explanation of summation notion as described in the text.\" width=\"731\" height=\"73\" \/> Figure 1. Labels for summation notation[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>If we interpret the given notation, we see that it asks us to find the sum of the terms in the series [latex]{a}_{k}=2k[\/latex] for [latex]k=1[\/latex] through [latex]k=5[\/latex]. We can begin by substituting the terms for [latex]k[\/latex] and listing out the terms of this series.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;{a}_{1}=2\\left(1\\right)=2 \\\\ &amp;{a}_{2}=2\\left(2\\right)=4 \\\\ &amp;{a}_{3}=2\\left(3\\right)=6 \\\\ &amp;{a}_{4}=2\\left(4\\right)=8 \\\\ &amp;{a}_{5}=2\\left(5\\right)=10 \\end{align}[\/latex]<\/p>\r\n<p>We can find the sum of the series by adding the terms:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\sum\\limits _{k=1}^{5}2k=2+4+6+8+10=30[\/latex]<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>summation notation<\/h3>\r\n<p>The sum of the first [latex]n[\/latex] terms of a <strong>series <\/strong>can be expressed in <strong>summation notation<\/strong> as follows:<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]\\sum\\limits _{k=1}^{n}{a}_{k}[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>This notation tells us to find the sum of [latex]{a}_{k}[\/latex] from:<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]k=1[\/latex] to [latex]k=n[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>[latex]k[\/latex] is called the <strong>index of summation<\/strong>, [latex]1[\/latex] is the <strong>lower limit of summation<\/strong>, and [latex]n[\/latex] is the <strong>upper limit of summation<\/strong>.<\/p>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p><strong>Q &amp; A: Does the lower limit of summation have to be 1?<\/strong><\/p>\r\n<p><em>No. The lower limit of summation can be any number, but 1 is frequently used. We will look at examples with lower limits of summation other than 1.<\/em><\/p>\r\n<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Given summation notation for a series, evaluate the value.<\/strong><\/p>\r\n<ol>\r\n\t<li>Identify the lower limit of summation.<\/li>\r\n\t<li>Identify the upper limit of summation.<\/li>\r\n\t<li>Substitute each value of [latex]k[\/latex] from the lower limit to the upper limit into the formula.<\/li>\r\n\t<li>Add to find the sum.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Evaluate [latex]\\sum\\limits _{k=3}^{7}{k}^{2}[\/latex]<\/p>\r\n<p>[reveal-answer q=\"991305\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"991305\"]<\/p>\r\n<p>According to the notation, the lower limit of summation is [latex]3[\/latex] and the upper limit is [latex]7[\/latex]. So we need to find the sum of [latex]{k}^{2}[\/latex] from [latex]k=3[\/latex] to [latex]k=7[\/latex]. We find the terms of the series by substituting [latex]k=3\\text{,}4\\text{,}5\\text{,}6[\/latex], and [latex]7[\/latex] into the function [latex]{k}^{2}[\/latex]. We add the terms to find the sum.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sum _{k=3}^{7}{k}^{2} &amp; ={3}^{2}+{4}^{2}+{5}^{2}+{6}^{2}+{7}^{2} \\\\ &amp; =9+16+25+36+49 \\\\ \\\\ &amp; =135 \\end{align}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]13920[\/ohm2_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Understand the basics of summations, including their notation, properties, and practical uses&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:5057,&quot;3&quot;:{&quot;1&quot;:0},&quot;9&quot;:0,&quot;10&quot;:0,&quot;11&quot;:3,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Understand the basics of summations, including their notation, properties, and practical uses<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Using Summation Notation<\/h2>\n<p>To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a <strong>series<\/strong>. Consider, for example, the following series.<\/p>\n<p style=\"text-align: center;\">[latex]3+7+11+15+19+\\cdots[\/latex]<\/p>\n<p>The <strong>[latex]n\\text{th }[\/latex] partial sum<\/strong> of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{S}_{n}\\text{ represents the partial sum.} \\\\ &{S}_{1}=3 \\\\ &{S}_{2}=3+7=10 \\\\ &{S}_{3}=3+7+11=21 \\\\ &{S}_{4}=3+7+11+15=36\\end{align}[\/latex]<\/p>\n<p><strong>Summation notation <\/strong>is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter <strong>sigma<\/strong>, [latex]\\Sigma[\/latex], to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma.<\/p>\n<ul>\n<li>A variable called the <strong>index of summation <\/strong>is written below the sigma.<\/li>\n<li>The index of summation is set equal to the <strong>lower limit of summation<\/strong>, which is the number used to generate the first term in the series.<\/li>\n<li>The number above the sigma, called the <strong>upper limit of summation<\/strong>, is the number used to generate the last term in a series.<\/li>\n<\/ul>\n<div style=\"text-align: center;\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03225157\/CNX_Precalc_Figure_11_04_001n2.jpg\" alt=\"Explanation of summation notion as described in the text.\" width=\"731\" height=\"73\" \/><figcaption class=\"wp-caption-text\">Figure 1. Labels for summation notation<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>If we interpret the given notation, we see that it asks us to find the sum of the terms in the series [latex]{a}_{k}=2k[\/latex] for [latex]k=1[\/latex] through [latex]k=5[\/latex]. We can begin by substituting the terms for [latex]k[\/latex] and listing out the terms of this series.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &{a}_{1}=2\\left(1\\right)=2 \\\\ &{a}_{2}=2\\left(2\\right)=4 \\\\ &{a}_{3}=2\\left(3\\right)=6 \\\\ &{a}_{4}=2\\left(4\\right)=8 \\\\ &{a}_{5}=2\\left(5\\right)=10 \\end{align}[\/latex]<\/p>\n<p>We can find the sum of the series by adding the terms:<\/p>\n<p style=\"text-align: center;\">[latex]\\sum\\limits _{k=1}^{5}2k=2+4+6+8+10=30[\/latex]<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>summation notation<\/h3>\n<p>The sum of the first [latex]n[\/latex] terms of a <strong>series <\/strong>can be expressed in <strong>summation notation<\/strong> as follows:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\sum\\limits _{k=1}^{n}{a}_{k}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>This notation tells us to find the sum of [latex]{a}_{k}[\/latex] from:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]k=1[\/latex] to [latex]k=n[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]k[\/latex] is called the <strong>index of summation<\/strong>, [latex]1[\/latex] is the <strong>lower limit of summation<\/strong>, and [latex]n[\/latex] is the <strong>upper limit of summation<\/strong>.<\/p>\n<\/section>\n<section class=\"textbox proTip\">\n<p><strong>Q &amp; A: Does the lower limit of summation have to be 1?<\/strong><\/p>\n<p><em>No. The lower limit of summation can be any number, but 1 is frequently used. We will look at examples with lower limits of summation other than 1.<\/em><\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Given summation notation for a series, evaluate the value.<\/strong><\/p>\n<ol>\n<li>Identify the lower limit of summation.<\/li>\n<li>Identify the upper limit of summation.<\/li>\n<li>Substitute each value of [latex]k[\/latex] from the lower limit to the upper limit into the formula.<\/li>\n<li>Add to find the sum.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate [latex]\\sum\\limits _{k=3}^{7}{k}^{2}[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q991305\">Show Solution<\/button><\/p>\n<div id=\"q991305\" class=\"hidden-answer\" style=\"display: none\">\n<p>According to the notation, the lower limit of summation is [latex]3[\/latex] and the upper limit is [latex]7[\/latex]. So we need to find the sum of [latex]{k}^{2}[\/latex] from [latex]k=3[\/latex] to [latex]k=7[\/latex]. We find the terms of the series by substituting [latex]k=3\\text{,}4\\text{,}5\\text{,}6[\/latex], and [latex]7[\/latex] into the function [latex]{k}^{2}[\/latex]. We add the terms to find the sum.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sum _{k=3}^{7}{k}^{2} & ={3}^{2}+{4}^{2}+{5}^{2}+{6}^{2}+{7}^{2} \\\\ & =9+16+25+36+49 \\\\ \\\\ & =135 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13920\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13920&theme=lumen&iframe_resize_id=ohm13920&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":92,"module-header":"background_you_need","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\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8348"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8348\/revisions"}],"predecessor-version":[{"id":15944,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8348\/revisions\/15944"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/92"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8348\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8348"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8348"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8348"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8348"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}