{"id":1678,"date":"2025-07-25T19:12:04","date_gmt":"2025-07-25T19:12:04","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1678"},"modified":"2026-03-25T21:47:42","modified_gmt":"2026-03-25T21:47:42","slug":"arithmetic-sequences-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/arithmetic-sequences-learn-it-3\/","title":{"raw":"Arithmetic Sequences: Learn It 3","rendered":"Arithmetic Sequences: Learn It 3"},"content":{"raw":"

Using Explicit Formulas for Arithmetic Sequences<\/h2>\r\nNow that you've written the terms of the sequence, your next task is to write the formula that represents the arithmetic sequence.\r\n\r\n
\r\n

explicit formula for an arithmetic sequence<\/h3>\r\nAn explicit formula for the [latex]n\\text{th}[\/latex] term of an arithmetic sequence is given by\r\n

[latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/p>\r\n\r\n<\/section>Try to use the pattern you\u2019ve identified in the terms to create a formula that can generate any term in the sequence.\r\n\r\n

How To: Given the first several terms for an arithmetic sequence, write an explicit formula.<\/strong>\r\n
    \r\n \t
  1. Find the common difference, [latex]{a}_{2}-{a}_{1}[\/latex].<\/li>\r\n \t
  2. 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<\/section>
    Write an explicit formula ([latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]) for the arithmetic sequence:\r\n
    [latex]\\left\\{2\\text{, }12\\text{, }22\\text{, }32\\text{, }42\\text{, }\\ldots \\right\\}[\/latex]<\/center>[reveal-answer q=\"533579\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"533579\"]The common difference can be found by subtracting the first term from the second term.\r\n

    [latex]\\begin{align}d&={a}_{2}-{a}_{1} \\\\ & =12 - 2 \\\\ & =10 \\end{align}[\/latex]<\/p>\r\nThe common difference is [latex]10[\/latex]. Substitute the common difference and the first term of the sequence into the formula and simplify.\r\n

    [latex]\\begin{align}&{a}_{n}=2+10\\left(n - 1\\right) \\\\ &{a}_{n}=10n - 8 \\end{align}[\/latex]<\/p>\r\n\"Graph\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nThe graph of this sequence shows a slope of [latex]10[\/latex] and a vertical intercept of [latex]-8[\/latex] .\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]321962[\/ohm_question]<\/section>\r\n

    Using Recursive Formulas for Arithmetic Sequences<\/h2>\r\nNow that you've written the explicit formula, let's explore a different approach: the recursive formula. Try writing a formula that defines each term of the sequence based on the previous term. Remember, the recursive formula builds the sequence step by step, starting from the first term.\r\n\r\n
    \r\n

    recursive formula for an arithmetic sequence<\/h3>\r\nThe recursive formula for an arithmetic sequence with common difference [latex]d[\/latex] is:\r\n

    [latex]\\begin{align}&{a}_{n}={a}_{n - 1}+d && n\\ge 2 \\end{align}[\/latex]<\/p>\r\n\r\n<\/section>

    How To: Given an arithmetic sequence, write its recursive formula.<\/strong>\r\n
      \r\n \t
    1. Subtract any term from the subsequent term to find the common difference.<\/li>\r\n \t
    2. State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.<\/li>\r\n<\/ol>\r\n<\/section>
      Write a recursive formula for the\u00a0arithmetic sequence.\r\n
      [latex]\\left\\{-18,-7,4,15,26, \\ldots \\right\\}[\/latex]<\/center>[reveal-answer q=\"265289\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"265289\"]The first term is given as [latex]-18[\/latex] . The common difference can be found by subtracting the first term from the second term.\r\n

      [latex]d=-7-\\left(-18\\right)=11[\/latex]<\/p>\r\nSubstitute the initial term and the common difference into the recursive formula for arithmetic sequences.\r\n

      [latex]\\begin{align}&{a}_{1}=-18 \\\\ &{a}_{n}={a}_{n - 1}+11,\\text{ for }n\\ge 2 \\end{align}[\/latex]<\/p>\r\n\"Graph\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nWe see that the common difference is the slope of the line formed when we graph the terms of the sequence. The growth pattern of the sequence shows the constant difference of [latex]11[\/latex] units.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

      [ohm_question hide_question_numbers=1]321963[\/ohm_question]<\/section>","rendered":"

      Using Explicit Formulas for Arithmetic Sequences<\/h2>\n

      Now that you’ve written the terms of the sequence, your next task is to write the formula that represents the arithmetic sequence.<\/p>\n

      \n

      explicit formula for an arithmetic sequence<\/h3>\n

      An explicit formula for the [latex]n\\text{th}[\/latex] term of an arithmetic sequence is given by<\/p>\n

      [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]<\/p>\n<\/section>\n

      Try to use the pattern you\u2019ve identified in the terms to create a formula that can generate any term in the sequence.<\/p>\n

      How To: Given the first several terms for an arithmetic sequence, write an explicit formula.<\/strong><\/p>\n
        \n
      1. Find the common difference, [latex]{a}_{2}-{a}_{1}[\/latex].<\/li>\n
      2. Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex].<\/li>\n<\/ol>\n<\/section>\n
        Write an explicit formula ([latex]{a}_{n}={a}_{1}+d\\left(n - 1\\right)[\/latex]) for the arithmetic sequence:<\/p>\n
        [latex]\\left\\{2\\text{, }12\\text{, }22\\text{, }32\\text{, }42\\text{, }\\ldots \\right\\}[\/latex]<\/div>\n