{"id":2496,"date":"2016-11-03T22:56:40","date_gmt":"2016-11-03T22:56:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=2496"},"modified":"2017-04-12T17:03:00","modified_gmt":"2017-04-12T17:03:00","slug":"arithmetic-series","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/arithmetic-series\/","title":{"raw":"Arithmetic Series","rendered":"Arithmetic Series"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Evaluate a sum given in summation notation<\/li>\r\n \t<li>Find the partial sum\u00a0of an arithmetic series<\/li>\r\n \t<li>Solve an application problem using an arithmetic series<\/li>\r\n<\/ul>\r\n<\/div>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a>\r\n\r\n&nbsp;\r\n<h2>Using Summation Notation<\/h2>\r\nTo 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.\r\n<p style=\"text-align: center;\">[latex]3+7+11+15+19+\\cdots[\/latex]<\/p>\r\nThe <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\r\n<p style=\"text-align: center;\">[latex]\\text{ }{S}_{n}\\text{ }[\/latex] represents the partial sum.\r\n[latex]\\begin{array}{l}{S}_{1}=3\\\\ {S}_{2}=3+7=10\\\\ {S}_{3}=3+7+11=21\\\\ {S}_{4}=3+7+11+15=36\\end{array}[\/latex]<\/p>\r\n<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. A variable called the <strong>index of summation <\/strong>is written below the sigma. 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. 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.\r\n\r\n<img class=\"aligncenter\" 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.\" data-media-type=\"image\/jpg\" \/>\r\n\r\nIf 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.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {a}_{1}=2\\left(1\\right)=2\\end{array}\\hfill \\\\ {a}_{2}=2\\left(2\\right)=4\\hfill \\\\ {a}_{3}=2\\left(3\\right)=6\\hfill \\\\ {a}_{4}=2\\left(4\\right)=8\\hfill \\\\ {a}_{5}=2\\left(5\\right)=10\\hfill \\end{array}[\/latex]<\/p>\r\nWe can find the sum of the series by adding the terms:\r\n<p style=\"text-align: center;\">[latex]\\sum _{k=1}^{5}2k=2+4+6+8+10=30[\/latex]<\/p>\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: Summation Notation<\/h3>\r\nThe sum of the first [latex]n[\/latex] terms of a <strong>series <\/strong>can be expressed in <strong>summation notation<\/strong> as follows:\r\n<p style=\"text-align: center;\">[latex]\\sum _{k=1}^{n}{a}_{k}[\/latex]<\/p>\r\nThis notation tells us to find the sum of [latex]{a}_{k}[\/latex] from\r\n<p style=\"text-align: center;\">[latex]k=1[\/latex] to [latex]k=n[\/latex].<\/p>\r\n[latex]k[\/latex] is called the <strong>index of summation<\/strong>, 1 is the <strong>lower limit of summation<\/strong>, and [latex]n[\/latex] is the <strong>upper limit of summation<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<h4>Does the lower limit of summation have to be 1?<\/h4>\r\n<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>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given summation notation for a series, evaluate the value.<\/h3>\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<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Summation Notation<\/h3>\r\nEvaluate [latex]\\sum _{k=3}^{7}{k}^{2}[\/latex].\r\n\r\n[reveal-answer q=\"991305\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"991305\"]\r\n\r\nAccording to the notation, the lower limit of summation is 3 and the upper limit is 7. 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.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}\\sum _{k=3}^{7}{k}^{2}\\hfill &amp; ={3}^{2}+{4}^{2}+{5}^{2}+{6}^{2}+{7}^{2}\\hfill \\\\ \\hfill &amp; =9+16+25+36+49\\hfill \\\\ \\hfill &amp; =135\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nEvaluate [latex]\\sum _{k=2}^{5}\\left(3k - 1\\right)[\/latex].\r\n\r\n[reveal-answer q=\"788016\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"788016\"]38[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=5866&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe>\r\n\r\n<\/div>\r\n<h3>Arithmetic Series<\/h3>\r\nJust as we studied special types of sequences, we will look at special types of series. Recall that an <strong>arithmetic sequence<\/strong> is a sequence in which the difference between any two consecutive terms is the <strong>common difference<\/strong>, [latex]d[\/latex]. The sum of the terms of an arithmetic sequence is called an <strong>arithmetic series<\/strong>. We can write the sum of the first [latex]n[\/latex] terms of an arithmetic series as:\r\n<p style=\"text-align: center;\">[latex]{S}_{n}={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+...+\\left({a}_{n}-d\\right)+{a}_{n}[\/latex].<\/p>\r\nWe can also reverse the order of the terms and write the sum as\r\n<p style=\"text-align: center;\">[latex]{S}_{n}={a}_{n}+\\left({a}_{n}-d\\right)+\\left({a}_{n}-2d\\right)+...+\\left({a}_{1}+d\\right)+{a}_{1}[\/latex].<\/p>\r\nIf we add these two expressions for the sum of the first [latex]n[\/latex] terms of an arithmetic series, we can derive a formula for the sum of the first [latex]n[\/latex] terms of any arithmetic series.\r\n<p style=\"text-align: center;\">[latex]\\frac{\\begin{array}{l}{S}_{n}={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+...+\\left({a}_{n}-d\\right)+{a}_{n}\\hfill \\\\ +{S}_{n}={a}_{n}+\\left({a}_{n}-d\\right)+\\left({a}_{n}-2d\\right)+...+\\left({a}_{1}+d\\right)+{a}_{1}\\hfill \\end{array}}{2{S}_{n}=\\left({a}_{1}+{a}_{n}\\right)+\\left({a}_{1}+{a}_{n}\\right)+...+\\left({a}_{1}+{a}_{n}\\right)}[\/latex]<\/p>\r\nBecause there are [latex]n[\/latex] terms in the series, we can simplify this sum to\r\n<p style=\"text-align: center;\">[latex]2{S}_{n}=n\\left({a}_{1}+{a}_{n}\\right)[\/latex].<\/p>\r\nWe divide by 2 to find the formula for the sum of the first [latex]n[\/latex] terms of an arithmetic series.\r\n<p style=\"text-align: center;\">[latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">This is generally referred to as the <strong>Partial Sum<\/strong> of the series.<\/p>\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: Formula for the Partial Sum of\u00a0an Arithmetic Series<\/h3>\r\nAn <strong>arithmetic series<\/strong> is the sum of the terms of an arithmetic sequence. The formula for the partial sum of an arithmetic sequence is\r\n<p style=\"text-align: center;\">[latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given terms of an arithmetic series, find the partial sum<\/h3>\r\n<ol>\r\n \t<li>Identify [latex]{a}_{1}[\/latex] and [latex]{a}_{n}[\/latex].<\/li>\r\n \t<li>Determine [latex]n[\/latex].<\/li>\r\n \t<li>Substitute values for [latex]{a}_{1}\\text{, }{a}_{n}[\/latex], and [latex]n[\/latex] into the formula [latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex].<\/li>\r\n \t<li>Simplify to find [latex]{S}_{n}[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the partial sum\u00a0of an Arithmetic Series<\/h3>\r\nFind the partial sum of each arithmetic series.\r\n<ol>\r\n \t<li>[latex]\\text{5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32}[\/latex]<\/li>\r\n \t<li>[latex]\\text{20 + 15 + 10 +\\ldots + -50}[\/latex]<\/li>\r\n \t<li>[latex]\\sum _{k=1}^{12}3k - 8[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"470866\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"470866\"]\r\n<ol>\r\n \t<li>We are given [latex]{a}_{1}=5[\/latex] and [latex]{a}_{n}=32[\/latex].Count the number of terms in the sequence to find [latex]n=10[\/latex].Substitute values for [latex]{a}_{1},{a}_{n}\\text{\\hspace{0.17em},}[\/latex] and [latex]n[\/latex] into the formula and simplify.\r\n[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ {S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\hfill \\end{array}\\hfill \\\\ {S}_{10}=\\frac{10\\left(5+32\\right)}{2}=185\\hfill \\end{array}[\/latex]<\/li>\r\n \t<li>We are given [latex]{a}_{1}=20[\/latex] and [latex]{a}_{n}=-50[\/latex].Use the formula for the general term of an arithmetic sequence to find [latex]n[\/latex].\r\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}+\\left(n - 1\\right)d\\hfill \\\\ -50=20+\\left(n - 1\\right)\\left(-5\\right)\\hfill \\\\ -70=\\left(n - 1\\right)\\left(-5\\right)\\hfill \\\\ 14=n - 1\\hfill \\\\ 15=n\\hfill \\end{array}[\/latex]\r\nSubstitute values for [latex]{a}_{1},{a}_{n}\\text{,}n[\/latex] into the formula and simplify.[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\end{array}\\hfill \\\\ {S}_{15}=\\frac{15\\left(20 - 50\\right)}{2}=-225\\hfill \\end{array}[\/latex]<\/li>\r\n \t<li>To find [latex]{a}_{1}[\/latex], substitute [latex]k=1[\/latex] into the given explicit formula.\r\n[latex]\\begin{array}{l}{a}_{k}=3k - 8\\hfill \\\\ \\text{ }{a}_{1}=3\\left(1\\right)-8=-5\\hfill \\end{array}[\/latex]\r\nWe are given that [latex]n=12[\/latex]. To find [latex]{a}_{12}[\/latex], substitute [latex]k=12[\/latex] into the given explicit formula.\r\n[latex]\\begin{array}{l}\\text{ }{a}_{k}=3k - 8\\hfill \\\\ {a}_{12}=3\\left(12\\right)-8=28\\hfill \\end{array}[\/latex]\r\nSubstitute values for [latex]{a}_{1},{a}_{n}[\/latex], and [latex]n[\/latex] into the formula and simplify.\r\n[latex]\\begin{array}{l}\\text{ }{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\hfill \\\\ {S}_{12}=\\frac{12\\left(-5+28\\right)}{2}=138\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse the formula to find the partial sum of each arithmetic series.\r\n\r\n[latex]\\text{1}\\text{.4 + 1}\\text{.6 + 1}\\text{.8 + 2}\\text{.0 + 2}\\text{.2 + 2}\\text{.4 + 2}\\text{.6 + 2}\\text{.8 + 3}\\text{.0 + 3}\\text{.2 + 3}\\text{.4}[\/latex]\r\n\r\n[reveal-answer q=\"649728\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"649728\"][latex]\\text{26}\\text{.4}[\/latex][\/hidden-answer]\r\n\r\n[latex]\\text{13 + 21 + 29 + }\\dots \\text{+ 69}[\/latex]\r\n\r\n[reveal-answer q=\"617640\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"617640\"][latex]\\text{328}[\/latex][\/hidden-answer]\r\n\r\n[latex]\\sum _{k=1}^{10}5 - 6k[\/latex]\r\n\r\n[reveal-answer q=\"794771\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"794771\"][latex]\\text{-280}[\/latex][\/hidden-answer]\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=128790&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe>\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=128791&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving Application Problems with Arithmetic Series<\/h3>\r\nOn the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?\r\n\r\n[reveal-answer q=\"455757\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"455757\"]\r\n\r\nThis problem can be modeled by an arithmetic series with [latex]{a}_{1}=\\frac{1}{2}[\/latex] and [latex]d=\\frac{1}{4}[\/latex]. We are looking for the total number of miles walked after 8 weeks, so we know that [latex]n=8[\/latex], and we are looking for [latex]{S}_{8}[\/latex]. To find [latex]{a}_{8}[\/latex], we can use the explicit formula for an arithmetic sequence.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {a}_{n}={a}_{1}+d\\left(n - 1\\right)\\end{array}\\hfill \\\\ {a}_{8}=\\frac{1}{2}+\\frac{1}{4}\\left(8 - 1\\right)=\\frac{9}{4}\\hfill \\end{array}[\/latex]<\/p>\r\nWe can now use the formula for arithmetic series.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} {S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\hfill \\\\ \\text{ }{S}_{8}=\\frac{8\\left(\\frac{1}{2}+\\frac{9}{4}\\right)}{2}=11\\hfill \\end{array}[\/latex]<\/p>\r\nShe will have walked a total of 11 miles.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nA man earns $100 in the first week of June. Each week, he earns $12.50 more than the previous week. After 12 weeks, how much has he earned?\r\n\r\n[reveal-answer q=\"454197\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"454197\"]$2,025[\/hidden-answer]\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=5865&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Evaluate a sum given in summation notation<\/li>\n<li>Find the partial sum\u00a0of an arithmetic series<\/li>\n<li>Solve an application problem using an arithmetic series<\/li>\n<\/ul>\n<\/div>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\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]\\text{ }{S}_{n}\\text{ }[\/latex] represents the partial sum.<br \/>\n[latex]\\begin{array}{l}{S}_{1}=3\\\\ {S}_{2}=3+7=10\\\\ {S}_{3}=3+7+11=21\\\\ {S}_{4}=3+7+11+15=36\\end{array}[\/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. A variable called the <strong>index of summation <\/strong>is written below the sigma. 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. 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.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" 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.\" data-media-type=\"image\/jpg\" \/><\/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{array}{l}\\begin{array}{l}\\\\ {a}_{1}=2\\left(1\\right)=2\\end{array}\\hfill \\\\ {a}_{2}=2\\left(2\\right)=4\\hfill \\\\ {a}_{3}=2\\left(3\\right)=6\\hfill \\\\ {a}_{4}=2\\left(4\\right)=8\\hfill \\\\ {a}_{5}=2\\left(5\\right)=10\\hfill \\end{array}[\/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 _{k=1}^{5}2k=2+4+6+8+10=30[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>A General Note: 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 style=\"text-align: center;\">[latex]\\sum _{k=1}^{n}{a}_{k}[\/latex]<\/p>\n<p>This notation tells us to find the sum of [latex]{a}_{k}[\/latex] from<\/p>\n<p style=\"text-align: center;\">[latex]k=1[\/latex] to [latex]k=n[\/latex].<\/p>\n<p>[latex]k[\/latex] is called the <strong>index of summation<\/strong>, 1 is the <strong>lower limit of summation<\/strong>, and [latex]n[\/latex] is the <strong>upper limit of summation<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>Does the lower limit of summation have to be 1?<\/h4>\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<\/div>\n<div class=\"textbox\">\n<h3>How To: Given summation notation for a series, evaluate the value.<\/h3>\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<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Summation Notation<\/h3>\n<p>Evaluate [latex]\\sum _{k=3}^{7}{k}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q991305\">Solution<\/span><\/p>\n<div id=\"q991305\" class=\"hidden-answer\" style=\"display: none\">\n<p>According to the notation, the lower limit of summation is 3 and the upper limit is 7. 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{array}{ll}\\sum _{k=3}^{7}{k}^{2}\\hfill & ={3}^{2}+{4}^{2}+{5}^{2}+{6}^{2}+{7}^{2}\\hfill \\\\ \\hfill & =9+16+25+36+49\\hfill \\\\ \\hfill & =135\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Evaluate [latex]\\sum _{k=2}^{5}\\left(3k - 1\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q788016\">Solution<\/span><\/p>\n<div id=\"q788016\" class=\"hidden-answer\" style=\"display: none\">38<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=5866&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<h3>Arithmetic Series<\/h3>\n<p>Just as we studied special types of sequences, we will look at special types of series. Recall that an <strong>arithmetic sequence<\/strong> is a sequence in which the difference between any two consecutive terms is the <strong>common difference<\/strong>, [latex]d[\/latex]. The sum of the terms of an arithmetic sequence is called an <strong>arithmetic series<\/strong>. We can write the sum of the first [latex]n[\/latex] terms of an arithmetic series as:<\/p>\n<p style=\"text-align: center;\">[latex]{S}_{n}={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+...+\\left({a}_{n}-d\\right)+{a}_{n}[\/latex].<\/p>\n<p>We can also reverse the order of the terms and write the sum as<\/p>\n<p style=\"text-align: center;\">[latex]{S}_{n}={a}_{n}+\\left({a}_{n}-d\\right)+\\left({a}_{n}-2d\\right)+...+\\left({a}_{1}+d\\right)+{a}_{1}[\/latex].<\/p>\n<p>If we add these two expressions for the sum of the first [latex]n[\/latex] terms of an arithmetic series, we can derive a formula for the sum of the first [latex]n[\/latex] terms of any arithmetic series.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\begin{array}{l}{S}_{n}={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+...+\\left({a}_{n}-d\\right)+{a}_{n}\\hfill \\\\ +{S}_{n}={a}_{n}+\\left({a}_{n}-d\\right)+\\left({a}_{n}-2d\\right)+...+\\left({a}_{1}+d\\right)+{a}_{1}\\hfill \\end{array}}{2{S}_{n}=\\left({a}_{1}+{a}_{n}\\right)+\\left({a}_{1}+{a}_{n}\\right)+...+\\left({a}_{1}+{a}_{n}\\right)}[\/latex]<\/p>\n<p>Because there are [latex]n[\/latex] terms in the series, we can simplify this sum to<\/p>\n<p style=\"text-align: center;\">[latex]2{S}_{n}=n\\left({a}_{1}+{a}_{n}\\right)[\/latex].<\/p>\n<p>We divide by 2 to find the formula for the sum of the first [latex]n[\/latex] terms of an arithmetic series.<\/p>\n<p style=\"text-align: center;\">[latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/p>\n<p style=\"text-align: left;\">This is generally referred to as the <strong>Partial Sum<\/strong> of the series.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Formula for the Partial Sum of\u00a0an Arithmetic Series<\/h3>\n<p>An <strong>arithmetic series<\/strong> is the sum of the terms of an arithmetic sequence. The formula for the partial sum of an arithmetic sequence is<\/p>\n<p style=\"text-align: center;\">[latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given terms of an arithmetic series, find the partial sum<\/h3>\n<ol>\n<li>Identify [latex]{a}_{1}[\/latex] and [latex]{a}_{n}[\/latex].<\/li>\n<li>Determine [latex]n[\/latex].<\/li>\n<li>Substitute values for [latex]{a}_{1}\\text{, }{a}_{n}[\/latex], and [latex]n[\/latex] into the formula [latex]{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex].<\/li>\n<li>Simplify to find [latex]{S}_{n}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the partial sum\u00a0of an Arithmetic Series<\/h3>\n<p>Find the partial sum of each arithmetic series.<\/p>\n<ol>\n<li>[latex]\\text{5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32}[\/latex]<\/li>\n<li>[latex]\\text{20 + 15 + 10 +\\ldots + -50}[\/latex]<\/li>\n<li>[latex]\\sum _{k=1}^{12}3k - 8[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q470866\">Solution<\/span><\/p>\n<div id=\"q470866\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>We are given [latex]{a}_{1}=5[\/latex] and [latex]{a}_{n}=32[\/latex].Count the number of terms in the sequence to find [latex]n=10[\/latex].Substitute values for [latex]{a}_{1},{a}_{n}\\text{\\hspace{0.17em},}[\/latex] and [latex]n[\/latex] into the formula and simplify.<br \/>\n[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ {S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\hfill \\end{array}\\hfill \\\\ {S}_{10}=\\frac{10\\left(5+32\\right)}{2}=185\\hfill \\end{array}[\/latex]<\/li>\n<li>We are given [latex]{a}_{1}=20[\/latex] and [latex]{a}_{n}=-50[\/latex].Use the formula for the general term of an arithmetic sequence to find [latex]n[\/latex].<br \/>\n[latex]\\begin{array}{l}{a}_{n}={a}_{1}+\\left(n - 1\\right)d\\hfill \\\\ -50=20+\\left(n - 1\\right)\\left(-5\\right)\\hfill \\\\ -70=\\left(n - 1\\right)\\left(-5\\right)\\hfill \\\\ 14=n - 1\\hfill \\\\ 15=n\\hfill \\end{array}[\/latex]<br \/>\nSubstitute values for [latex]{a}_{1},{a}_{n}\\text{,}n[\/latex] into the formula and simplify.[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\end{array}\\hfill \\\\ {S}_{15}=\\frac{15\\left(20 - 50\\right)}{2}=-225\\hfill \\end{array}[\/latex]<\/li>\n<li>To find [latex]{a}_{1}[\/latex], substitute [latex]k=1[\/latex] into the given explicit formula.<br \/>\n[latex]\\begin{array}{l}{a}_{k}=3k - 8\\hfill \\\\ \\text{ }{a}_{1}=3\\left(1\\right)-8=-5\\hfill \\end{array}[\/latex]<br \/>\nWe are given that [latex]n=12[\/latex]. To find [latex]{a}_{12}[\/latex], substitute [latex]k=12[\/latex] into the given explicit formula.<br \/>\n[latex]\\begin{array}{l}\\text{ }{a}_{k}=3k - 8\\hfill \\\\ {a}_{12}=3\\left(12\\right)-8=28\\hfill \\end{array}[\/latex]<br \/>\nSubstitute values for [latex]{a}_{1},{a}_{n}[\/latex], and [latex]n[\/latex] into the formula and simplify.<br \/>\n[latex]\\begin{array}{l}\\text{ }{S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\hfill \\\\ {S}_{12}=\\frac{12\\left(-5+28\\right)}{2}=138\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use the formula to find the partial sum of each arithmetic series.<\/p>\n<p>[latex]\\text{1}\\text{.4 + 1}\\text{.6 + 1}\\text{.8 + 2}\\text{.0 + 2}\\text{.2 + 2}\\text{.4 + 2}\\text{.6 + 2}\\text{.8 + 3}\\text{.0 + 3}\\text{.2 + 3}\\text{.4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q649728\">Solution<\/span><\/p>\n<div id=\"q649728\" class=\"hidden-answer\" style=\"display: none\">[latex]\\text{26}\\text{.4}[\/latex]<\/div>\n<\/div>\n<p>[latex]\\text{13 + 21 + 29 + }\\dots \\text{+ 69}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q617640\">Solution<\/span><\/p>\n<div id=\"q617640\" class=\"hidden-answer\" style=\"display: none\">[latex]\\text{328}[\/latex]<\/div>\n<\/div>\n<p>[latex]\\sum _{k=1}^{10}5 - 6k[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q794771\">Solution<\/span><\/p>\n<div id=\"q794771\" class=\"hidden-answer\" style=\"display: none\">[latex]\\text{-280}[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=128790&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=128791&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Application Problems with Arithmetic Series<\/h3>\n<p>On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q455757\">Solution<\/span><\/p>\n<div id=\"q455757\" class=\"hidden-answer\" style=\"display: none\">\n<p>This problem can be modeled by an arithmetic series with [latex]{a}_{1}=\\frac{1}{2}[\/latex] and [latex]d=\\frac{1}{4}[\/latex]. We are looking for the total number of miles walked after 8 weeks, so we know that [latex]n=8[\/latex], and we are looking for [latex]{S}_{8}[\/latex]. To find [latex]{a}_{8}[\/latex], we can use the explicit formula for an arithmetic sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {a}_{n}={a}_{1}+d\\left(n - 1\\right)\\end{array}\\hfill \\\\ {a}_{8}=\\frac{1}{2}+\\frac{1}{4}\\left(8 - 1\\right)=\\frac{9}{4}\\hfill \\end{array}[\/latex]<\/p>\n<p>We can now use the formula for arithmetic series.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l} {S}_{n}=\\frac{n\\left({a}_{1}+{a}_{n}\\right)}{2}\\hfill \\\\ \\text{ }{S}_{8}=\\frac{8\\left(\\frac{1}{2}+\\frac{9}{4}\\right)}{2}=11\\hfill \\end{array}[\/latex]<\/p>\n<p>She will have walked a total of 11 miles.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A man earns $100 in the first week of June. Each week, he earns $12.50 more than the previous week. After 12 weeks, how much has he earned?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q454197\">Solution<\/span><\/p>\n<div id=\"q454197\" class=\"hidden-answer\" style=\"display: none\">$2,025<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=5865&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2496\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 5865, 5867. <strong>Authored by<\/strong>: WebWork-Rochester. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>:  IMathAS Community License CC-BY +GPL<\/li><li>Question ID 128790,128791. <strong>Authored by<\/strong>: Day, Alyson. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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GPL\"}]","CANDELA_OUTCOMES_GUID":"baedc370-22d0-4a1b-9807-c4a65022fdf2","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2496","chapter","type-chapter","status-publish","hentry"],"part":2419,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2496","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2496\/revisions"}],"predecessor-version":[{"id":4100,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2496\/revisions\/4100"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/2419"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2496\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/media?parent=2496"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2496"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/contributor?post=2496"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/license?post=2496"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}