Example  1
Determine if the following infinite sequences are geometric. Justify your decisions.

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

    In order for the sequence to be considered geometric, there must be a common ratio. Calculate \(r \) for pairs of consecutive terms.
    \[\begin{align}
     r &= \frac{t_n }{t_{n - 1}} \\
     r& = \frac{6}{2} = 3 \\
     r &= \frac{18}{6} = 3 \\
     r &= \frac{54}{18} = 3 \\
     \end{align}\]

    Because \(r \) is constant, this is a geometric sequence.

  2. \(1, 2, 6, 24, ...\)

    Calculate \(r \) for pairs of consecutive terms.
    \[\begin{align}
     r &= \frac{t_n }{t_{n - 1}} \\
     r &= \frac{2}{1} = 2 \\
     r &= \frac{6}{2} = 3 \\
     \end{align} \]
    You can stop at this point because there is no common ratio; therefore, the sequence is not geometric.

  3. \(100, 50, 25, 12.5, ...\)

    Calculate \(r \) for pairs of consecutive terms.
    \[\begin{align}
     r &= \frac{t_n }{t_{n - 1}} \\
     r &= \frac{50}{100} = 0.5 \\
     r &= \frac{25}{50} = 0.5 \\
     r &= \frac{12.5}{25} = 0.5 \\
     \end{align} \]

    Because \(r \) is constant, this is a geometric sequence.

  4. \(\frac{1}{7}, -\frac{1}{49}, \frac{1}{343}, -\frac{1}{2401}, ...\)

    Calculate \(r \) for pairs of consecutive terms.
    \[\begin{align}
     r &= \frac{{t_n }}{{t_{n - 1} }} \\
     r &= \frac{{ -\frac{1}{{49}}}}{{\frac{1}{7}}} = -\frac{1}{7} \\
     r &= \frac{{\frac{1}{{343}}}}{{ - \frac{1}{{49}}}} = -\frac{1}{7} \\
     r &= \frac{{ -\frac{1}{{2{\rm{ }}401}}}}{{\frac{1}{{343}}}} = -\frac{1}{7} \\
     \end{align} \]
    Because \(r \) is constant, this is a geometric sequence.