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Determine the sums of the following geometric series.

  1. \(1 + 5 + 25 + 125 + ... + t_6 \)

    Use a geometric series formula to find the sum of the series.
    \[\begin{align}
    t_1 &= 1 \\
    r &= \frac{5}{1} = 5 \\
    n &= 6 \\
    S_6 &= ? \\
    \end{align}\]
    \[\begin{align}
     S_n &= \frac{{t_1 \left( {r^n - 1} \right)}}{{r - 1}} \\
     S_6 &= \frac{{1\left( {5^6 - 1} \right)}}{{5 - 1}} \\
     S_6 &= \frac{{15\thinspace 624}}{4} \\
     S_6 &= 3\thinspace 906 \\
     \end{align}\]
    The sum of the series is \(3\thinspace 906\).

  2. \(S_8 = 16 + 4 + 1 + \frac{1}{4} + ... \)

    Use a geometric series formula to find the sum of the series.

    \[\begin{align}
    t_1 &= 16 \\
    r &= \frac{4}{16} = \frac{1}{4} \\
    n &= 8 \\
    S_8 &= ? \\
    \end{align}\]

    \[\begin{align}
     S_n &= \frac{t_1(r^n - 1)}{r - 1} \\
     S_8 &= \frac{16\left [{\frac{1}{4}}^8 - 1\right ]}{\frac{1}{4} - 1} \\
     S_8 &= \frac{-15.999...}{-\frac{3}{4}} \\
     S_8 &= 21.333... \\
     S_8 &= \frac{21\thinspace 845}{1\thinspace 024} \\
     \end{align}\]

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  3. \(t_n = 32, t_1 = \frac{1}{16}, r = -2 \)

    Because \(n \) is not given, you can use the second formula.

    \(\begin{align}
     S_n &= \frac{{rt_n - t_1 }}{{r - 1}} \\
     S_n &= \frac{{\left( {-2} \right)\left( {32} \right) - \left( {\frac{1}{{16}}} \right)}}{{-2 - 1}} \\
     S_n &= \frac{{\frac{{-1\thinspace 025}}{{16}}}}{{-3}} \\
     S_n &= 21.354... \\
     S_n &= \frac{{1\thinspace 025}}{{48}} \\
     \end{align}\)