{"id":14790,"date":"2018-09-27T18:36:01","date_gmt":"2018-09-27T18:36:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/arithmetic-sequences-2\/"},"modified":"2021-06-30T14:35:55","modified_gmt":"2021-06-30T14:35:55","slug":"arithmetic-sequences-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/chapter\/arithmetic-sequences-2\/","title":{"raw":"Arithmetic Sequences","rendered":"Arithmetic Sequences"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Find the common difference for an arithmetic sequence.<\/li>\r\n \t<li style=\"font-weight: 400;\">Give terms of an arithmetic sequence.<\/li>\r\n \t<li style=\"font-weight: 400;\">Write the formula for an arithmetic sequence.<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe sequence {25000, 21600, 18200, 1480, 11400, 8000} are the values of a truck each year.\u00a0 The values of the truck are said to form an <strong>arithmetic sequence<\/strong> because they change by a constant amount each year. Each term increases or decreases by the same constant value called the <strong>common difference<\/strong> of the sequence. For this sequence, the common difference is \u20133,400.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183536\/CNX_Precalc_Figure_11_02_0012.jpg\" alt=\"A sequence, {25000, 21600, 18200, 14800, 8000}, that shows the terms differ only by -3400.\" \/>\r\n\r\nThe sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can choose any <strong>term<\/strong> of the <strong>sequence<\/strong>, and add 3 to find the subsequent term.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183538\/CNX_Precalc_Figure_11_02_0022.jpg\" alt=\"A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Arithmetic Sequence<\/h3>\r\nAn <strong>arithmetic sequence<\/strong> is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the <strong>common difference<\/strong>. If [latex]{a}_{1}[\/latex] is the first term of an arithmetic sequence and [latex]d[\/latex] is the common difference, the sequence will be:\r\n<p style=\"text-align: center;\">[latex]\\left\\{{a}_{n}\\right\\}=\\left\\{{a}_{1},{a}_{1}+d,{a}_{1}+2d,{a}_{1}+3d,...\\right\\}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 1: Finding Common Differences<\/h3>\r\nIs each sequence arithmetic? If so, find the common difference.\r\n<ol>\r\n \t<li>[latex]\\left\\{1,2,4,8,16,...\\right\\}[\/latex]<\/li>\r\n \t<li>[latex]\\left\\{-3,1,5,9,13,...\\right\\}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"815179\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"815179\"]\r\n\r\nSubtract each term from the subsequent term to determine whether a common difference exists.\r\n<ol>\r\n \t<li>The sequence is not arithmetic because there is no common difference.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183540\/Eqn12.jpg\" alt=\"2 minus 1 = 1. 4 minus 2 = 2. 8 minus 4 = 4. 16 minus 8 equals 8.\" width=\"475\" height=\"27\" \/><\/li>\r\n \t<li>The sequence is arithmetic because there is a common difference. The common difference is 4.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183542\/Eqn22.jpg\" alt=\"1 minus negative 3 equals 4. 5 minus 1 equals 4. 9 minus 5 equals 4. 13 minus 9 equals 4.\" width=\"505\" height=\"27\" \/><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nIs the given sequence arithmetic? If so, find the common difference.\r\n<p style=\"text-align: center;\">[latex]\\left\\{18,\\text{ }16,\\text{ }14,\\text{ }12,\\text{ }10,\\dots \\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"157343\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"157343\"]\r\n\r\nThe sequence is arithmetic. The common difference is [latex]-2[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nIs the given sequence arithmetic? If so, find the common difference.\r\n<p style=\"text-align: center;\">[latex]\\left\\{1,\\text{ }3,\\text{ }6,\\text{ }10,\\text{ }15,\\dots \\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"151399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"151399\"]\r\n\r\nThe sequence is not arithmetic because [latex]3 - 1\\ne 6 - 3[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Writing Terms of Arithmetic Sequences<\/h2>\r\nNow that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of [latex]n[\/latex] and [latex]d[\/latex] into formula below.\r\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the first term and the common difference of an arithmetic sequence, find the first several terms.<\/h3>\r\n<ol>\r\n \t<li>Add the common difference to the first term to find the second term.<\/li>\r\n \t<li>Add the common difference to the second term to find the third term.<\/li>\r\n \t<li>Continue until all of the desired terms are identified.<\/li>\r\n \t<li>Write the terms separated by commas within brackets.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 2: Writing Terms of Arithmetic Sequences<\/h3>\r\nWrite the first five terms of the <strong>arithmetic sequence<\/strong> with [latex]{a}_{1}=17[\/latex] and [latex]d=-3[\/latex].\r\n\r\n[reveal-answer q=\"223260\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"223260\"]\r\n\r\nAdding [latex]-3[\/latex] is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term.\r\n\r\nThe first five terms are [latex]\\left\\{17,14,11,8,5\\right\\}[\/latex]\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">[\/hidden-answer]<\/span>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nList the first five terms of the arithmetic sequence with [latex]{a}_{1}=1[\/latex] and [latex]d=5[\/latex] .\r\n\r\n[reveal-answer q=\"531419\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"531419\"]\r\n\r\n[latex]\\left\\{1, 6, 11, 16, 21\\right\\}[\/latex]\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\n[ohm_question hide_question_numbers=1]79151[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the first term and any other term in an arithmetic sequence, find a given term.<\/h3>\r\n<ol>\r\n \t<li>Substitute the values given for [latex]{a}_{1},{a}_{n},n[\/latex] into the formula [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex] to solve for [latex]d[\/latex].<\/li>\r\n \t<li>Find a given term by substituting the appropriate values for [latex]{a}_{1},n[\/latex], and [latex]d[\/latex] into the formula [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 3: Writing Terms of Arithmetic Sequences<\/h3>\r\nGiven [latex]{a}_{1}=8[\/latex] and [latex]{a}_{4}=14[\/latex] , find [latex]{a}_{5}[\/latex].\r\n\r\n[reveal-answer q=\"473118\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"473118\"]\r\n\r\nThe sequence can be written in terms of the initial term 8 and the common difference [latex]d[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\left\\{8,8+d,8+2d,8+3d,\\dots\\right\\}[\/latex]<\/p>\r\nWe know the fourth term equals 14; we know the fourth term has the form [latex]{a}_{1}+3d=8+3d[\/latex].\r\n\r\nWe can find the common difference [latex]d[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{n}={a}_{1}+\\left(n - 1\\right)d \\\\ &amp;{a}_{4}={a}_{1}+3d \\\\ &amp;{a}_{4}=8+3d&amp;&amp; \\text{Write the fourth term of the sequence in terms of } {a}_{1} \\text{ and } d. \\\\ &amp;14=8+3d&amp;&amp; \\text{Substitute } 14 \\text{ for } {a}_{4}. \\\\ &amp;d=2&amp;&amp; \\text{Solve for the common difference}. \\end{align}[\/latex]<\/p>\r\nFind the fifth term by adding the common difference to the fourth term.\r\n<p style=\"text-align: center;\">[latex]{a}_{5}={a}_{4}+2=16[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nGiven [latex]{a}_{3}=7[\/latex] and [latex]{a}_{5}=17[\/latex] , find [latex]{a}_{2}[\/latex] .\r\n\r\n[reveal-answer q=\"797232\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"797232\"]\r\n\r\n[latex]{a}_{2}=2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the first several terms for an arithmetic sequence, write an explicit formula.<\/h3>\r\n<ol>\r\n \t<li>Find the common difference, [latex]{a}_{2}-{a}_{1}[\/latex].<\/li>\r\n \t<li>Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 5: Writing the <em>n<\/em>th Term Explicit Formula for an Arithmetic Sequence<\/h3>\r\nWrite an explicit formula for the arithmetic sequence.\r\n<p style=\"text-align: center;\">[latex]\\left\\{2\\text{, }12\\text{, }22\\text{, }32\\text{, }42\\text{, }\\dots \\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"360705\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"360705\"]\r\n\r\nThe common difference can be found by subtracting the first term from the second term.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}d\\hfill &amp; ={a}_{2}-{a}_{1} \\\\ &amp; =12 - 2 \\\\ &amp;=10 \\end{align}[\/latex]<\/p>\r\nThe common difference is 10. Substitute the common difference and the first term of the sequence into the formula and simplify.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{n}=2+10\\left(n - 1\\right) \\\\ &amp;{a}_{n}=10n - 8 \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nWrite an explicit formula for the following arithmetic sequence.\r\n<p style=\"text-align: center;\">[latex]\\left\\{50,47,44,41,\\dots \\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"42399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"42399\"]\r\n\r\n[latex]{a}_{n}=53 - 3n[\/latex]\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\n[ohm_question hide_question_numbers=1]172280[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Finding the Number of Terms in a Finite Arithmetic Sequence<\/h2>\r\nExplicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.\r\n<div class=\"textbox\">\r\n<h3>How To: Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.<\/h3>\r\n<ol>\r\n \t<li>Find the common difference [latex]d[\/latex].<\/li>\r\n \t<li>Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\r\n \t<li>Substitute the last term for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 6: Finding the Number of Terms in a Finite Arithmetic Sequence<\/h3>\r\nFind the number of terms in the <strong>finite arithmetic sequence<\/strong>.\r\n<p style=\"text-align: center;\">[latex]\\left\\{8\\text{, }1\\text{, }-6\\text{, }\\dots\\text{, }-41\\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"672670\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"672670\"]\r\n\r\nThe common difference can be found by subtracting the first term from the second term.\r\n<p style=\"text-align: center;\">[latex]1 - 8=-7[\/latex]<\/p>\r\nThe common difference is [latex]-7[\/latex] . Substitute the common difference and the initial term of the sequence into the [latex]n\\text{th}[\/latex] term formula and simplify.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{a}_{n}={a}_{1}+d\\left(n - 1\\right)\\\\ &amp;{a}_{n}=8+-7\\left(n - 1\\right)\\\\ &amp;{a}_{n}=15 - 7n \\end{align}[\/latex]<\/p>\r\nSubstitute [latex]-41[\/latex] for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}-41&amp;=15 - 7n \\\\ 8&amp;=n \\end{align}[\/latex]<\/p>\r\nThere are eight terms in the sequence.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nFind the number of terms in the finite arithmetic sequence.\r\n<p style=\"text-align: center;\">[latex]\\left\\{6,11,16,\\dots,56\\right\\}[\/latex]<\/p>\r\n[reveal-answer q=\"12633\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"12633\"]\r\n\r\nThere are 11 terms in the sequence.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\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<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{0}+dn[\/latex]<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 7: Solving Application Problems with Arithmetic Sequences<\/h3>\r\nA five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 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 16 years old?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"502685\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"502685\"]\r\n<ol>\r\n \t<li>The situation can be modeled by an arithmetic sequence with an initial term of 1 and a common difference of 2.Let [latex]A[\/latex] be the amount of the allowance and [latex]n[\/latex] be the number of years after age 5. Using the altered explicit formula for an arithmetic sequence we get:\r\n<div style=\"text-align: center;\">[latex]{A}_{n}=1+2n[\/latex]<\/div><\/li>\r\n \t<li>We can find the number of years since age 5 by subtracting.\r\n<div style=\"text-align: center;\">[latex]16 - 5=11[\/latex]<\/div>\r\nWe are looking for the child\u2019s allowance after 11 years. Substitute 11 into the formula to find the child\u2019s allowance at age 16.\r\n<div style=\"text-align: center;\">[latex]{A}_{11}=1+2\\left(11\\right)=23[\/latex]<\/div>\r\nThe child\u2019s allowance at age 16 will be $23 per week.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nA woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from today?\r\n\r\n[reveal-answer q=\"433471\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"433471\"]\r\n\r\nThe formula is [latex]{T}_{n}=10+4n[\/latex], and it will take her 42 minutes.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li style=\"font-weight: 400;\">Find the common difference for an arithmetic sequence.<\/li>\n<li style=\"font-weight: 400;\">Give terms of an arithmetic sequence.<\/li>\n<li style=\"font-weight: 400;\">Write the formula for an arithmetic sequence.<\/li>\n<\/ul>\n<\/div>\n<p>The sequence {25000, 21600, 18200, 1480, 11400, 8000} are the values of a truck each year.\u00a0 The values of the truck are said to form an <strong>arithmetic sequence<\/strong> because they change by a constant amount each year. Each term increases or decreases by the same constant value called the <strong>common difference<\/strong> of the sequence. For this sequence, the common difference is \u20133,400.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183536\/CNX_Precalc_Figure_11_02_0012.jpg\" alt=\"A sequence, {25000, 21600, 18200, 14800, 8000}, that shows the terms differ only by -3400.\" \/><\/p>\n<p>The sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can choose any <strong>term<\/strong> of the <strong>sequence<\/strong>, and add 3 to find the subsequent term.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183538\/CNX_Precalc_Figure_11_02_0022.jpg\" alt=\"A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Arithmetic Sequence<\/h3>\n<p>An <strong>arithmetic sequence<\/strong> is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the <strong>common difference<\/strong>. If [latex]{a}_{1}[\/latex] is the first term of an arithmetic sequence and [latex]d[\/latex] is the common difference, the sequence will be:<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{{a}_{n}\\right\\}=\\left\\{{a}_{1},{a}_{1}+d,{a}_{1}+2d,{a}_{1}+3d,...\\right\\}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Finding Common Differences<\/h3>\n<p>Is each sequence arithmetic? If so, find the common difference.<\/p>\n<ol>\n<li>[latex]\\left\\{1,2,4,8,16,...\\right\\}[\/latex]<\/li>\n<li>[latex]\\left\\{-3,1,5,9,13,...\\right\\}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q815179\">Show Solution<\/span><\/p>\n<div id=\"q815179\" class=\"hidden-answer\" style=\"display: none\">\n<p>Subtract each term from the subsequent term to determine whether a common difference exists.<\/p>\n<ol>\n<li>The sequence is not arithmetic because there is no common difference.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183540\/Eqn12.jpg\" alt=\"2 minus 1 = 1. 4 minus 2 = 2. 8 minus 4 = 4. 16 minus 8 equals 8.\" width=\"475\" height=\"27\" \/><\/li>\n<li>The sequence is arithmetic because there is a common difference. The common difference is 4.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183542\/Eqn22.jpg\" alt=\"1 minus negative 3 equals 4. 5 minus 1 equals 4. 9 minus 5 equals 4. 13 minus 9 equals 4.\" width=\"505\" height=\"27\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Is the given sequence arithmetic? If so, find the common difference.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{18,\\text{ }16,\\text{ }14,\\text{ }12,\\text{ }10,\\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q157343\">Show Solution<\/span><\/p>\n<div id=\"q157343\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence is arithmetic. The common difference is [latex]-2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Is the given sequence arithmetic? If so, find the common difference.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{1,\\text{ }3,\\text{ }6,\\text{ }10,\\text{ }15,\\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q151399\">Show Solution<\/span><\/p>\n<div id=\"q151399\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence is not arithmetic because [latex]3 - 1\\ne 6 - 3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Writing Terms of Arithmetic Sequences<\/h2>\n<p>Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of [latex]n[\/latex] and [latex]d[\/latex] into formula below.<\/p>\n<div style=\"text-align: center;\">[latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the first term and the common difference of an arithmetic sequence, find the first several terms.<\/h3>\n<ol>\n<li>Add the common difference to the first term to find the second term.<\/li>\n<li>Add the common difference to the second term to find the third term.<\/li>\n<li>Continue until all of the desired terms are identified.<\/li>\n<li>Write the terms separated by commas within brackets.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Writing Terms of Arithmetic Sequences<\/h3>\n<p>Write the first five terms of the <strong>arithmetic sequence<\/strong> with [latex]{a}_{1}=17[\/latex] and [latex]d=-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q223260\">Show Solution<\/span><\/p>\n<div id=\"q223260\" class=\"hidden-answer\" style=\"display: none\">\n<p>Adding [latex]-3[\/latex] is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term.<\/p>\n<p>The first five terms are [latex]\\left\\{17,14,11,8,5\\right\\}[\/latex]<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>List the first five terms of the arithmetic sequence with [latex]{a}_{1}=1[\/latex] and [latex]d=5[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q531419\">Show Solution<\/span><\/p>\n<div id=\"q531419\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left\\{1, 6, 11, 16, 21\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm79151\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=79151&theme=oea&iframe_resize_id=ohm79151\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the first term and any other term in an arithmetic sequence, find a given term.<\/h3>\n<ol>\n<li>Substitute the values given for [latex]{a}_{1},{a}_{n},n[\/latex] into the formula [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex] to solve for [latex]d[\/latex].<\/li>\n<li>Find a given term by substituting the appropriate values for [latex]{a}_{1},n[\/latex], and [latex]d[\/latex] into the formula [latex]{a}_{n}={a}_{1}+\\left(n - 1\\right)d[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Writing Terms of Arithmetic Sequences<\/h3>\n<p>Given [latex]{a}_{1}=8[\/latex] and [latex]{a}_{4}=14[\/latex] , find [latex]{a}_{5}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q473118\">Show Solution<\/span><\/p>\n<div id=\"q473118\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sequence can be written in terms of the initial term 8 and the common difference [latex]d[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{8,8+d,8+2d,8+3d,\\dots\\right\\}[\/latex]<\/p>\n<p>We know the fourth term equals 14; we know the fourth term has the form [latex]{a}_{1}+3d=8+3d[\/latex].<\/p>\n<p>We can find the common difference [latex]d[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{a}_{n}={a}_{1}+\\left(n - 1\\right)d \\\\ &{a}_{4}={a}_{1}+3d \\\\ &{a}_{4}=8+3d&& \\text{Write the fourth term of the sequence in terms of } {a}_{1} \\text{ and } d. \\\\ &14=8+3d&& \\text{Substitute } 14 \\text{ for } {a}_{4}. \\\\ &d=2&& \\text{Solve for the common difference}. \\end{align}[\/latex]<\/p>\n<p>Find the fifth term by adding the common difference to the fourth term.<\/p>\n<p style=\"text-align: center;\">[latex]{a}_{5}={a}_{4}+2=16[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Given [latex]{a}_{3}=7[\/latex] and [latex]{a}_{5}=17[\/latex] , find [latex]{a}_{2}[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q797232\">Show Solution<\/span><\/p>\n<div id=\"q797232\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{2}=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the first several terms for an arithmetic sequence, write an explicit formula.<\/h3>\n<ol>\n<li>Find the common difference, [latex]{a}_{2}-{a}_{1}[\/latex].<\/li>\n<li>Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 5: Writing the <em>n<\/em>th Term Explicit Formula for an Arithmetic Sequence<\/h3>\n<p>Write an explicit formula for the arithmetic sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{2\\text{, }12\\text{, }22\\text{, }32\\text{, }42\\text{, }\\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q360705\">Show Solution<\/span><\/p>\n<div id=\"q360705\" class=\"hidden-answer\" style=\"display: none\">\n<p>The common difference can be found by subtracting the first term from the second term.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}d\\hfill & ={a}_{2}-{a}_{1} \\\\ & =12 - 2 \\\\ &=10 \\end{align}[\/latex]<\/p>\n<p>The common difference is 10. Substitute the common difference and the first term of the sequence into the formula and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{a}_{n}=2+10\\left(n - 1\\right) \\\\ &{a}_{n}=10n - 8 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Write an explicit formula for the following arithmetic sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{50,47,44,41,\\dots \\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q42399\">Show Solution<\/span><\/p>\n<div id=\"q42399\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{a}_{n}=53 - 3n[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm172280\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=172280&theme=oea&iframe_resize_id=ohm172280\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Finding the Number of Terms in a Finite Arithmetic Sequence<\/h2>\n<p>Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.<\/h3>\n<ol>\n<li>Find the common difference [latex]d[\/latex].<\/li>\n<li>Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\n<li>Substitute the last term for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 6: Finding the Number of Terms in a Finite Arithmetic Sequence<\/h3>\n<p>Find the number of terms in the <strong>finite arithmetic sequence<\/strong>.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{8\\text{, }1\\text{, }-6\\text{, }\\dots\\text{, }-41\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q672670\">Show Solution<\/span><\/p>\n<div id=\"q672670\" class=\"hidden-answer\" style=\"display: none\">\n<p>The common difference can be found by subtracting the first term from the second term.<\/p>\n<p style=\"text-align: center;\">[latex]1 - 8=-7[\/latex]<\/p>\n<p>The common difference is [latex]-7[\/latex] . Substitute the common difference and the initial term of the sequence into the [latex]n\\text{th}[\/latex] term formula and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{a}_{n}={a}_{1}+d\\left(n - 1\\right)\\\\ &{a}_{n}=8+-7\\left(n - 1\\right)\\\\ &{a}_{n}=15 - 7n \\end{align}[\/latex]<\/p>\n<p>Substitute [latex]-41[\/latex] for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}-41&=15 - 7n \\\\ 8&=n \\end{align}[\/latex]<\/p>\n<p>There are eight terms in the sequence.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the number of terms in the finite arithmetic sequence.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{6,11,16,\\dots,56\\right\\}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q12633\">Show Solution<\/span><\/p>\n<div id=\"q12633\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are 11 terms in the sequence.<\/p>\n<\/div>\n<\/div>\n<\/div>\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<div class=\"textbox shaded\">\n<h3>Example 7: Solving Application Problems with Arithmetic Sequences<\/h3>\n<p>A five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 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 16 years old?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q502685\">Show Solution<\/span><\/p>\n<div id=\"q502685\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The situation can be modeled by an arithmetic sequence with an initial term of 1 and a common difference of 2.Let [latex]A[\/latex] be the amount of the allowance and [latex]n[\/latex] be the number of years after age 5. 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 5 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 11 years. Substitute 11 into the formula to find the child\u2019s allowance at age 16.<\/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 16 will be $23 per week.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>A woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from today?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q433471\">Show Solution<\/span><\/p>\n<div id=\"q433471\" class=\"hidden-answer\" style=\"display: none\">\n<p>The formula is [latex]{T}_{n}=10+4n[\/latex], and it will take her 42 minutes.<\/p>\n<\/div>\n<\/div>\n<\/div>\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-14790\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-14790","chapter","type-chapter","status-publish","hentry"],"part":14758,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters\/14790","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":12,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters\/14790\/revisions"}],"predecessor-version":[{"id":16145,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters\/14790\/revisions\/16145"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/parts\/14758"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapters\/14790\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/media?parent=14790"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/pressbooks\/v2\/chapter-type?post=14790"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/contributor?post=14790"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nwfsc-MGF1107\/wp-json\/wp\/v2\/license?post=14790"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}