Example  2
Determine the general term for each of the geometric sequences in Example 1.

  1. \(2, 6, 18, 54, ...\)

    \(t_1 = 2 \) and \(r = 3 \), so the general term is \(t_n = 2\left( 3 \right)^{n - 1} \).
  1. \(100, 50, 25, 12.5, ...\)

    \(t_1 = 100 \) and \(r = 0.5 \) or \(\frac{1}{2} \), so the general term is \(t_n = 100\left( {0.5} \right)^{n - 1} \) or \(t_n = 100\left( {\frac{1}{2}} \right)^{n - 1}  \).
  2. \(\frac{1}{7}, -\frac{1}{49}, \frac{1}{343}, -\frac{1}{2401}, ...\)

    \(t_1 = \frac{1}{7} \) and \(r = -\frac{1}{7} \), so the general term is
    \[
    \begin{array}{l}
     t_n = \left( {\frac{1}{7}} \right)\left( { - \frac{1}{7}} \right)^{n - 1}  \\
     t_n = \left( {\frac{1}{7}} \right)\left( { -1} \right)^{n - 1} \left( {\frac{1}{7}} \right)^{n - 1}  \\
     t_n = \left( { -1} \right)^{n - 1} \left( {\frac{1}{7}} \right)^n  \\
     \end{array}
    \]