{"id":1673,"date":"2025-07-25T19:12:50","date_gmt":"2025-07-25T19:12:50","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1673"},"modified":"2026-03-26T16:02:19","modified_gmt":"2026-03-26T16:02:19","slug":"series-and-their-notations-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/series-and-their-notations-learn-it-2\/","title":{"raw":"Series and Their Notations: Learn It 2","rendered":"Series and Their Notations: Learn It 2"},"content":{"raw":"

Arithmetic Series<\/h2>\r\nJust as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference. The sum of the terms of an arithmetic sequence is called an arithmetic series<\/strong>.\r\n\r\n
\r\n

formula for the partial sum of an arithmetic series<\/h3>\r\nThe sum, [latex]S_n[\/latex], of the first [latex]n[\/latex] terms of an arithmetic sequence is<\/span>\r\n

[latex]S_n = \\sum_{k=1}^{n} a_k = \\dfrac{n}{2}(a_1 + a_n)[\/latex]<\/p>\r\nwhere [latex]a_1[\/latex] is the first term and [latex]a_n[\/latex] is the [latex]n[\/latex]th term.\r\n\r\n<\/section>[reveal-answer q=\"298285\"]Derivation of the formula.[\/reveal-answer]\r\n[hidden-answer a=\"298285\"]\r\n\r\nWe can write the sum of the first [latex]n[\/latex] terms of an arithmetic series as:\r\n

[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

[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

[latex]\\begin{align}{S}_{n}&={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+...+\\left({a}_{n}-d\\right)+{a}_{n} \\\\ +{S}_{n}&={a}_{n}+\\left({a}_{n}-d\\right)+\\left({a}_{n}-2d\\right)+...+\\left({a}_{1}+d\\right)+{a}_{1} \\\\ \\hline 2{S}_{n}&=\\left({a}_{1}+{a}_{n}\\right)+\\left({a}_{1}+{a}_{n}\\right)+...+\\left({a}_{1}+{a}_{n}\\right) \\end{align}[\/latex]<\/p>\r\nBecause there are [latex]n[\/latex] terms in the series, we can simplify this sum to\r\n

[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

[latex]{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex]<\/p>\r\n

This is generally referred to as the partial sum<\/strong> of the series.<\/p>\r\n[\/hidden-answer]\r\n\r\n

How To: Given terms of an arithmetic series, find the partial sum<\/strong>\r\n
    \r\n \t
  1. Identify [latex]{a}_{1}[\/latex] and [latex]{a}_{n}[\/latex].<\/li>\r\n \t
  2. Determine [latex]n[\/latex].<\/li>\r\n \t
  3. Substitute values for [latex]{a}_{1},{a}_{n}[\/latex], and [latex]n[\/latex] into the formula [latex]{S}_{n}=\\dfrac{n\\left({a}_{1}+{a}_{n}\\right)}{2}[\/latex].<\/li>\r\n \t
  4. Simplify to find [latex]{S}_{n}[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
    Find the sum of the first [latex]n[\/latex] terms of an arithmetic sequence.\r\n
      \r\n \t
    • Find the sum of the first [latex]30[\/latex] terms of the arithmetic sequence:
      [latex]7, 10, 13, 13, 19,...[\/latex]<\/center>\r\n[reveal-answer q=\"269302\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"269302\"][latex]\\begin{align*} \\text{To find the 30th term, use the formula with } a_1 = 7, \\, d = 3, \\text{ and } n = 30.\\end{align*}[\/latex]\r\n[latex]a_n = a_1 + (n - 1)d[\/latex]\r\n[latex]\\begin{align*} \\text{Substitute} \\quad & a_{30} = 7 + (30 - 1)(3) \\end{align*}[\/latex]\r\n[latex]\\begin{align*} \\text{Simplify} \\quad & a_{30} = 7 + (29)(8) \\\\ & a_{30} = 7 + 232 \\\\ & a_{30} = 239 \\end{align*}[\/latex]\r\n[latex]\\begin{align*} \\text{To find } S_{30} \\text{ use the formula with } a_1 = 7, \\, a_{30} = 239, \\text{ and } n = 30.\u00a0 \\end{align*}[\/latex]\r\n[latex]S_n = \\frac{n}{2}(a_1 + a_n)[\/latex]\r\n[latex]\\begin{align*} \\text{Substitute and simplify} \\quad & S_{30} = \\frac{30}{2} (7 + 239) \\\\ & S_{30} = 15(246) \\\\ & S_{30} = 3690 \\end{align*}[\/latex][\/hidden-answer]<\/li>\r\n \t
    • Find the sum of the first [latex]50[\/latex] terms of the arithmetic sequence whose general term is [latex]a_n = 2n-5[\/latex].\r\n[reveal-answer q=\"41989\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"41989\"][latex]\\begin{align*} \\text{To find the sum of the first 50 terms, we start by finding } a_1 \\text{ and } a_{50}. \\\\ \\text{The general term is given by } a_n &= 2n - 5. \\\\ \\text{First, find the first term } a_1: & \\\\ a_1 &= 2(1) - 5 \\\\ a_1 &= 2 - 5 \\\\ a_1 &= -3 \\\\ \\text{Next, find the 50th term } a_{50}: & \\\\ a_{50} &= 2(50) - 5 \\\\ a_{50} &= 100 - 5 \\\\ a_{50} &= 95 \\\\ \\text{Now, use the sum formula for an arithmetic sequence:} & \\\\ S_n &= \\frac{n}{2} \\left(a_1 + a_n\\right) \\\\ S_{50} &= \\frac{50}{2} \\left(-3 + 95\\right) \\\\ S_{50} &= 25 \\times 92 \\\\ S_{50} &= 2300 \\end{align*}[\/latex][\/hidden-answer]<\/li>\r\n \t
    • Find the sum
      [latex]\\sum_{i=1}^{30} (6i-4)[\/latex]<\/center>\r\n[reveal-answer q=\"59919\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"59919\"][latex]\\begin{align*} \\text{To find the sum } \\sum_{i=1}^{30} (6i-4), \\text{ first identify the sequence terms.} & \\\\ \\text{The first term is:} & \\\\ a_1 &= 6(1) - 4 \\\\ a_1 &= 6 - 4 \\\\ a_1 &= 2 \\\\ \\text{The 30th term is:} & \\\\ a_{30} &= 6(30) - 4 \\\\ a_{30} &= 180 - 4 \\\\ a_{30} &= 176 \\\\ \\text{Now, use the sum formula:} & \\\\ S_n &= \\frac{n}{2} \\left(a_1 + a_n\\right) \\\\ S_{30} &= \\frac{30}{2} \\left(2 + 176\\right) \\\\ S_{30} &= 15 \\times 178 \\\\ S_{30} &= 2670 \\end{align*}[\/latex][\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/section>
      [ohm_question hide_question_numbers=1]321979[\/ohm_question]<\/section>
      [ohm_question hide_question_numbers=1]321980[\/ohm_question]<\/section>","rendered":"

      Arithmetic Series<\/h2>\n

      Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference. The sum of the terms of an arithmetic sequence is called an arithmetic series<\/strong>.<\/p>\n

      \n

      formula for the partial sum of an arithmetic series<\/h3>\n

      The sum, [latex]S_n[\/latex], of the first [latex]n[\/latex] terms of an arithmetic sequence is<\/span><\/p>\n

      [latex]S_n = \\sum_{k=1}^{n} a_k = \\dfrac{n}{2}(a_1 + a_n)[\/latex]<\/p>\n

      where [latex]a_1[\/latex] is the first term and [latex]a_n[\/latex] is the [latex]n[\/latex]th term.<\/p>\n<\/section>\n