{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T15:10:37+00:00","modifiedTime":"2022-10-06T15:09:45+00:00","timestamp":"2022-10-06T18:01:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"Pre-Calculus: Finding the General Formula for the nth Term","strippedTitle":"pre-calculus: finding the general formula for the nth term","slug":"how-to-find-the-general-formula-for-the-nth-term-of-an-arithmetic-sequence-using-any-two-terms","canonicalUrl":"","seo":{"metaDescription":"Learn how to find the general formula for the nth term of an arithmetic sequence without knowing the first term or common difference.","noIndex":0,"noFollow":0},"content":"At some point, your pre-calculus teacher will ask you to find the general formula for the <i>n</i>th term of an arithmetic sequence without knowing the first term or the common difference. In this case, you will be given two terms (not necessarily consecutive), and you will use this information to find <i>a</i><sub>1</sub> and <i>d.</i> The steps are: Find the common difference <i>d</i>, write the specific formula for the given sequence, and then find the term you're looking for.\r\n\r\nFor instance, to find the general formula of an arithmetic sequence where <i>a</i><sub>4</sub> = –23 and <i>a</i><sub>22</sub> = 40, follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Find the common difference.</p>\r\n<p class=\"child-para\">You have to be creative in finding the common difference for these types of problems.</p>\r\n\r\n<ol class=\"level-two\">\r\n \t<li>\r\n<p class=\"first-para\">a.Use the formula <i>a</i><i><sub>n</sub></i> = <i>a</i><sub>1</sub> + (<i>n</i> – 1)<i>d</i> to set up two equations that use the given information.</p>\r\n<p class=\"child-para\">For the first equation, you know that when <i>n</i> = 4, <i>a</i><i><sub>n</sub></i> = –23:</p>\r\n<p class=\"child-para\">–23 = <i>a</i><sub>1</sub> + (4 – 1)<i>d</i></p>\r\n<p class=\"child-para\">–23 = <i>a</i><sub>1</sub> + 3<i>d</i></p>\r\n<p class=\"child-para\">For the second equation, you know that when <i>n</i> = 22, <i>a</i><i><sub>n</sub></i> = 40:</p>\r\n<p class=\"child-para\">40 = <i>a</i><sub>1</sub> + (22 – 1)<i>d</i></p>\r\n<p class=\"child-para\">40 = <i>a</i><sub>1</sub> + 21<i>d</i></p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">b.Set up a system of equations and solve for <i>d.</i></p>\r\n<p class=\"child-para\">The system looks like this:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370612.image0.png\" alt=\"image0.png\" width=\"199\" height=\"56\" />\r\n<p class=\"child-para\">You can use elimination or substitution to solve the system. Elimination works nicely because you can multiply either equation by –1 and add the two together to get 63 = 18<i>d. </i>Therefore, <i>d</i> = 3.5.</p>\r\n</li>\r\n</ol>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Write the formula for the specific sequence.</p>\r\n<p class=\"child-para\">This step involves a little work.</p>\r\n\r\n<ol class=\"level-two\">\r\n \t<li>\r\n<p class=\"first-para\">a.Plug <i>d</i> into one of the equations to solve for <i>a</i><sub>1</sub><i>.</i></p>\r\n<p class=\"child-para\">You can plug 3.5 back into either equation:</p>\r\n<p class=\"child-para\">–23 = <i>a</i><sub>1</sub> + 3(3.5), or <i>a</i><sub>1</sub> = –33.5.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">b.Use <i>a</i><sub>1</sub> and <i>d</i> to find the general formula for <i>a</i><i><sub>n</sub></i><i>.</i></p>\r\n<p class=\"child-para\">This step becomes a simple three-step simplification:</p>\r\n<p class=\"child-para\"><i>a</i><i><sub>n</sub></i><i> </i>= –33.5 + (<i>n </i>– 1)3.5</p>\r\n<p class=\"child-para\"><i>a</i><i><sub>n</sub></i> = –33.5 + 3.5<i>n</i> – 3.5</p>\r\n<p class=\"child-para\"><i>a</i><i><sub>n</sub></i><i> </i>= 3.5<i>n</i> – 37</p>\r\n</li>\r\n</ol>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Find the term you were looking for.</p>\r\n<p class=\"child-para\">In this example, you weren't asked to find any specific term (always read the directions!), but if you were, you could plug that number in for <i>n</i> and then find the term you were looking for.</p>\r\n</li>\r\n</ol>","description":"At some point, your pre-calculus teacher will ask you to find the general formula for the <i>n</i>th term of an arithmetic sequence without knowing the first term or the common difference. In this case, you will be given two terms (not necessarily consecutive), and you will use this information to find <i>a</i><sub>1</sub> and <i>d.</i> The steps are: Find the common difference <i>d</i>, write the specific formula for the given sequence, and then find the term you're looking for.\r\n\r\nFor instance, to find the general formula of an arithmetic sequence where <i>a</i><sub>4</sub> = –23 and <i>a</i><sub>22</sub> = 40, follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Find the common difference.</p>\r\n<p class=\"child-para\">You have to be creative in finding the common difference for these types of problems.</p>\r\n\r\n<ol class=\"level-two\">\r\n \t<li>\r\n<p class=\"first-para\">a.Use the formula <i>a</i><i><sub>n</sub></i> = <i>a</i><sub>1</sub> + (<i>n</i> – 1)<i>d</i> to set up two equations that use the given information.</p>\r\n<p class=\"child-para\">For the first equation, you know that when <i>n</i> = 4, <i>a</i><i><sub>n</sub></i> = –23:</p>\r\n<p class=\"child-para\">–23 = <i>a</i><sub>1</sub> + (4 – 1)<i>d</i></p>\r\n<p class=\"child-para\">–23 = <i>a</i><sub>1</sub> + 3<i>d</i></p>\r\n<p class=\"child-para\">For the second equation, you know that when <i>n</i> = 22, <i>a</i><i><sub>n</sub></i> = 40:</p>\r\n<p class=\"child-para\">40 = <i>a</i><sub>1</sub> + (22 – 1)<i>d</i></p>\r\n<p class=\"child-para\">40 = <i>a</i><sub>1</sub> + 21<i>d</i></p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">b.Set up a system of equations and solve for <i>d.</i></p>\r\n<p class=\"child-para\">The system looks like this:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370612.image0.png\" alt=\"image0.png\" width=\"199\" height=\"56\" />\r\n<p class=\"child-para\">You can use elimination or substitution to solve the system. Elimination works nicely because you can multiply either equation by –1 and add the two together to get 63 = 18<i>d. </i>Therefore, <i>d</i> = 3.5.</p>\r\n</li>\r\n</ol>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Write the formula for the specific sequence.</p>\r\n<p class=\"child-para\">This step involves a little work.</p>\r\n\r\n<ol class=\"level-two\">\r\n \t<li>\r\n<p class=\"first-para\">a.Plug <i>d</i> into one of the equations to solve for <i>a</i><sub>1</sub><i>.</i></p>\r\n<p class=\"child-para\">You can plug 3.5 back into either equation:</p>\r\n<p class=\"child-para\">–23 = <i>a</i><sub>1</sub> + 3(3.5), or <i>a</i><sub>1</sub> = –33.5.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">b.Use <i>a</i><sub>1</sub> and <i>d</i> to find the general formula for <i>a</i><i><sub>n</sub></i><i>.</i></p>\r\n<p class=\"child-para\">This step becomes a simple three-step simplification:</p>\r\n<p class=\"child-para\"><i>a</i><i><sub>n</sub></i><i> </i>= –33.5 + (<i>n </i>– 1)3.5</p>\r\n<p class=\"child-para\"><i>a</i><i><sub>n</sub></i> = –33.5 + 3.5<i>n</i> – 3.5</p>\r\n<p class=\"child-para\"><i>a</i><i><sub>n</sub></i><i> </i>= 3.5<i>n</i> – 37</p>\r\n</li>\r\n</ol>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Find the term you were looking for.</p>\r\n<p class=\"child-para\">In this example, you weren't asked to find any specific term (always read the directions!), but if you were, you could plug that number in for <i>n</i> and then find the term you were looking for.</p>\r\n</li>\r\n</ol>","blurb":"","authors":[{"authorId":9703,"name":"Yang Kuang","slug":"yang-kuang","description":"","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9703"}},{"authorId":9704,"name":"Elleyne Kase","slug":"elleyne-kase","description":"","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9704"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":[{"articleId":172026,"title":"Repurposing Items for Beautiful Holiday Decorating","slug":"decorating-for-the-holidays-with-what-you-already-have","categoryList":["home-auto-hobbies","crafts","gifts-holidays"],"_links":{"self":"/articles/172026"}},{"articleId":192609,"title":"How to Pray the Rosary: A Comprehensive Guide","slug":"how-to-pray-the-rosary","categoryList":["body-mind-spirit","religion-spirituality","christianity","catholicism"],"_links":{"self":"/articles/192609"}},{"articleId":193770,"title":"How 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She taught for more than 35 years and continues to teach via distance learning. Sterling is the author of many Dummies math titles, including <i>Pre-Calculus For Dummies</i>. She is a graduate of the University of New Hampshire with a master’s degree in mathematics education.</p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> was a professor of mathematics at Bradley University. She taught for more than 35 years and continues to teach via distance learning. Sterling is the author of many Dummies math titles, including <i>Pre-Calculus For Dummies</i>. She is a graduate of the University of New Hampshire with a master’s degree in mathematics education. 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Pre-Calculus: Finding the General Formula for the nth Term

By: Yang Kuang and Elleyne Kase and

Updated: 10-06-2022

Pre-Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)

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At some point, your pre-calculus teacher will ask you to find the general formula for the nth term of an arithmetic sequence without knowing the first term or the common difference. In this case, you will be given two terms (not necessarily consecutive), and you will use this information to find a₁ and d. The steps are: Find the common difference d, write the specific formula for the given sequence, and then find the term you're looking for.

For instance, to find the general formula of an arithmetic sequence where a₄ = –23 and a₂₂ = 40, follow these steps:

Find the common difference.

You have to be creative in finding the common difference for these types of problems.
1. a.Use the formula a_n = a₁ + (n – 1)d to set up two equations that use the given information.
  
  For the first equation, you know that when n = 4, a_n = –23:
  
  –23 = a₁ + (4 – 1)d
  
  –23 = a₁ + 3d
  
  For the second equation, you know that when n = 22, a_n = 40:
  
  40 = a₁ + (22 – 1)d
  
  40 = a₁ + 21d
2. b.Set up a system of equations and solve for d.
  
  The system looks like this:
  
  You can use elimination or substitution to solve the system. Elimination works nicely because you can multiply either equation by –1 and add the two together to get 63 = 18d. Therefore, d = 3.5.
Write the formula for the specific sequence.

This step involves a little work.
1. a.Plug d into one of the equations to solve for a₁.
  
  You can plug 3.5 back into either equation:
  
  –23 = a₁ + 3(3.5), or a₁ = –33.5.
2. b.Use a₁ and d to find the general formula for a_n.
  
  This step becomes a simple three-step simplification:
  
  a_n = –33.5 + (n – 1)3.5
  
  a_n = –33.5 + 3.5n – 3.5
  
  a_n = 3.5n – 37
Find the term you were looking for.

In this example, you weren't asked to find any specific term (always read the directions!), but if you were, you could plug that number in for n and then find the term you were looking for.

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