{"id":1672,"date":"2025-07-25T19:12:53","date_gmt":"2025-07-25T19:12:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1672"},"modified":"2026-03-26T16:04:11","modified_gmt":"2026-03-26T16:04:11","slug":"series-and-their-notations-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/series-and-their-notations-learn-it-3\/","title":{"raw":"Series and Their Notations: Learn It 3","rendered":"Series and Their Notations: Learn It 3"},"content":{"raw":"

Geometric Series<\/h2>\r\nJust as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series<\/strong>.\r\n\r\n
\r\n

formula for the sum of the first [latex]n[\/latex] terms of a geometric series<\/h3>\r\nA geometric series<\/strong> is the sum of the terms in a geometric sequence.\r\n\r\nThe formula for the sum of the first [latex]n[\/latex] terms of a geometric sequence is represented as\r\n

[latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\text{ r}\\ne \\text{1}[\/latex]<\/p>\r\n\r\n<\/section>[reveal-answer q=\"24709\"]Derivation of the formula.[\/reveal-answer]\r\n[hidden-answer a=\"24709\"]\r\n\r\nRecall that a geometric sequence<\/strong> is a sequence in which the ratio of any two consecutive terms is the common ratio<\/strong>, [latex]r[\/latex]. We can write the sum of the first [latex]n[\/latex] terms of a geometric series as\r\n

[latex]{S}_{n}={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+...+{a}_{1}{r}^{n - 1}[\/latex].<\/p>\r\nJust as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first [latex]n[\/latex] terms of a geometric series. We will begin by multiplying both sides of the equation by [latex]r[\/latex].\r\n

[latex]r{S}_{n}={a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+...+{a}_{1}{r}^{n}[\/latex]<\/p>\r\nNext, we subtract this equation from the original equation.\r\n

[latex]\\begin{align}{S}_{n}&={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+...+{a}_{1}{r}^{n - 1} \\\\ -r{S}_{n}&=-\\left({a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+...+{a}_{1}{r}^{n}\\right) \\\\ \\hline \\left(1-r\\right){S}_{n}&={a}_{1}-{a}_{1}{r}^{n}\\end{align}[\/latex]<\/p>\r\nNotice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for [latex]{S}_{n}[\/latex], factor [latex]a_1[\/latex] on the right hand side and divide both sides by [latex]\\left(1-r\\right)[\/latex].\r\n

[latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\text{ r}\\ne \\text{1}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n

How To: Given a geometric series, find the sum of the first [latex]n[\/latex] terms.<\/strong>\r\n
    \r\n \t
  1. Identify [latex]{a}_{1},r,\\text{ and }n[\/latex].<\/li>\r\n \t
  2. Substitute values for [latex]{a}_{1},r[\/latex], and [latex]n[\/latex] into the formula [latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}[\/latex].<\/li>\r\n \t
  3. Simplify to find [latex]{S}_{n}[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
    Use the formula to find the indicated partial sum of each geometric series.\r\n
      \r\n \t
    1. [latex]{S}_{11}[\/latex] for the series [latex] 8 + -4 + 2 + \\dots [\/latex]\r\n[reveal-answer q=\"373815\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"373815\"][latex]{a}_{1}=8[\/latex], and we are given that [latex]n=11[\/latex]. We can find [latex]r[\/latex] by dividing the second term of the series by the first.\r\n
      [latex]r=\\dfrac{-4}{8}=-\\frac{1}{2}[\/latex]<\/center>\r\nSubstitute values for [latex]{a}_{1}, r, \\text{ and } n[\/latex] into the formula and simplify.\r\n
      [latex]\\begin{align}&{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r} \\\\[1mm] &{S}_{11}=\\dfrac{8\\left(1-{\\left(-\\frac{1}{2}\\right)}^{11}\\right)}{1-\\left(-\\frac{1}{2}\\right)}\\approx 5.336 \\\\ \\text{ } \\end{align}[\/latex]<\/center>[\/hidden-answer]<\/li>\r\n \t
    2. [latex]\\sum\\limits _{k=1}^6 3\\cdot {2}^{k}[\/latex]\r\n[reveal-answer q=\"369192\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"369192\"]Find [latex]{a}_{1}[\/latex] by substituting [latex]k=1[\/latex] into the given explicit formula.\r\n
      [latex]{a}_{1}=3\\cdot {2}^{1}=6[\/latex]<\/center>\r\nWe can see from the given explicit formula that [latex]r=2[\/latex]. The upper limit of summation is [latex]6[\/latex], so [latex]n=6[\/latex].Substitute values for [latex]{a}_{1},r[\/latex], and [latex]n[\/latex] into the formula, and simplify.
      [latex]\\begin{align}\\\\ &{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r} \\\\[1mm] &{S}_{6}=\\frac{6\\left(1-{2}^{6}\\right)}{1 - 2}=378 \\end{align}[\/latex]<\/center>[\/hidden-answer]<\/li>\r\n<\/ol>\r\n \r\n\r\n<\/section>
      [ohm_question hide_question_numbers=1]321981[\/ohm_question]<\/section>
      [ohm_question hide_question_numbers=1]321982[\/ohm_question]<\/section>","rendered":"

      Geometric Series<\/h2>\n

      Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series<\/strong>.<\/p>\n

      \n

      formula for the sum of the first [latex]n[\/latex] terms of a geometric series<\/h3>\n

      A geometric series<\/strong> is the sum of the terms in a geometric sequence.<\/p>\n

      The formula for the sum of the first [latex]n[\/latex] terms of a geometric sequence is represented as<\/p>\n

      [latex]{S}_{n}=\\dfrac{{a}_{1}\\left(1-{r}^{n}\\right)}{1-r}\\text{ r}\\ne \\text{1}[\/latex]<\/p>\n<\/section>\n