{"id":1677,"date":"2025-07-25T19:12:08","date_gmt":"2025-07-25T19:12:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1677"},"modified":"2026-03-25T21:49:24","modified_gmt":"2026-03-25T21:49:24","slug":"arithmetic-sequences-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/arithmetic-sequences-learn-it-4\/","title":{"raw":"Arithmetic Sequences: Learn It 4","rendered":"Arithmetic Sequences: Learn It 4"},"content":{"raw":"

Find the Number of Terms in an 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\r\n
How To: Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.<\/strong>\r\n
    \r\n \t
  1. Find the common difference [latex]d[\/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 \t
  3. Substitute the last term for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
    Find the number of terms in the finite arithmetic sequence:\r\n

    [latex]\\left\\{8,1,-6, \\dots ,-41\\right\\}[\/latex]<\/p>\r\n\r\n

      \r\n \t
    1. The common difference can be found by subtracting the first term from the second term\r\n[latex]1 - 8=-7[\/latex]\r\nThe common difference is [latex]-7[\/latex].<\/li>\r\n \t
    2. Substitute the common difference and the initial term of the sequence into the [latex]n\\text{th}[\/latex] term formula and simplify.[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]<\/li>\r\n \t
    3. Substitute [latex]-41[\/latex] for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex]<\/li>\r\n<\/ol>\r\n

      [latex]\\begin{align}-41&=15 - 7n \\\\ 8&=n \\end{align}[\/latex]<\/p>\r\nThis means that there are eight terms<\/strong> in the sequence.\r\n\r\n<\/section>

      [ohm_question hide_question_numbers=1]321964[\/ohm_question]<\/section>","rendered":"

      Find the Number of Terms in an Arithmetic Sequence<\/h2>\n

      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

      How To: Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.<\/strong><\/p>\n
        \n
      1. Find the common difference [latex]d[\/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
      3. Substitute the last term for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex].<\/li>\n<\/ol>\n<\/section>\n
        Find the number of terms in the finite arithmetic sequence:<\/p>\n

        [latex]\\left\\{8,1,-6, \\dots ,-41\\right\\}[\/latex]<\/p>\n

          \n
        1. The common difference can be found by subtracting the first term from the second term
          \n[latex]1 - 8=-7[\/latex]
          \nThe common difference is [latex]-7[\/latex].<\/li>\n
        2. Substitute the common difference and the initial term of the sequence into the [latex]n\\text{th}[\/latex] term formula and simplify.[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]<\/li>\n
        3. Substitute [latex]-41[\/latex] for [latex]{a}_{n}[\/latex] and solve for [latex]n[\/latex]<\/li>\n<\/ol>\n

          [latex]\\begin{align}-41&=15 - 7n \\\\ 8&=n \\end{align}[\/latex]<\/p>\n

          This means that there are eight terms<\/strong> in the sequence.<\/p>\n<\/section>\n