Example 1
Determine the sums of the following geometric series.

  1. \(S_{12} = 2 + 6 + 18 + ... \)

    In this series, all necessary values are available to use the first formula.

    \[\begin{align}
    t_1 &= 2 \\
    r &= \frac{6}{2} = 3 \\
    n &= 12 \\
    S_{12} &= ? \\
    \end{align}\]
    \[\begin{align}
     S_n &= \frac{t_1(r^n - 1)}{r - 1} \\
     S_{12} &= \frac{2(3^{12} - 1)}{3 - 1} \\
     S_{12} &= \frac{1\thinspace 062\thinspace 880}{2} \\
     S_{12} &= 531\thinspace 440 \\
     \end{align}
    \]
    The sum of this series is \(531\thinspace 440\).
  2. \(135 + 45 + 15 + ... + \frac{5}{27}\)

    In this example, you are not given the value of \(n \), but you do have the value of \(t_n \). Use the second version of the formula. Alternatively, you can determine the value of \(n \) by using the general term formula for a geometric sequence, and then you can use the formula for the sum of a geometric series.

    \[\begin{align}
    t_1 &= 135 \\
    t_n &= \frac{5}{27} \\
    r &= \frac{45}{135} = \frac{1}{3} \\
    S_n &= ? \\
    \end{align}\]
    \[\begin{align}
     S_n &= \frac{rt_n - t_1}{r - 1} \\
     S_n &= \frac{{(\frac{1}{3})}{(\frac{5}{27})} - 135}{\frac{1}{3} - 1} \\
     S_n &= \frac{\frac{5}{81} - 135}{-\frac{2}{3}} \\
     S_n &= \frac{-\frac{{10\thinspace 930}}{{81}}}{{-\frac{2}{3}}} \\
     S_n &= 202.407... \\
     S_n &= \frac{5\thinspace 465}{27} \\
     \end{align}\]
    The sum of the series is \(\frac{5\thinspace 465}{27} \), or approximately \(202.41\).