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Assume that the following sequences are geometric. Find the missing terms, and write the general term.

  1.  \(405,\_\_\_,\_\_\_,15,\_\_\_\)

    \(\begin{align}
    t_1 &= 405 \\
    t_4 &= 15 \\
    n &= 4 \\
    r &= ? \\
    \end{align}\)
    \[\begin{align}
     t_n &= t_1 r^{n - 1}  \\
     15 &= 405r^{4 - 1}  \\
     \frac{1}{{27}} &= r^3  \\
     \sqrt[3]{{\frac{1}{{27}}}} &= r \\
     \frac{1}{3} &= r \\
     \end{align}\]


    The sequence is \(405, 135, 45, 15, 5\).

    The general term is \(t_n = t_1 r^{n - 1} = 405\left( {\frac{1}{3}} \right)^{n - 1} \).

  2. \(\_\_\_,\frac{1}{{16}},\_\_\_,\_\_\_,\frac{1}{{1\thinspace 024}}\)

    Equation I
    \[\begin{align}
    t_1 &= ? \\
    t_2 &= \frac{1}{16} \\
    n &= 2 \\
    r &= ? \\
    \end{align}\]
    \[\begin{align}
     t_n &= t_1 r^{n - 1}  \\
     \frac{1}{{16}} &= t_1 r^{2 - 1}  \\
     \frac{1}{{16}} &= t_1 r \\
     \frac{1}{{16r}} &= t_1  \\
     \end{align}\]
    Equation II
    \[\begin{align}
    t_1 &= ? \\
    t_5 &= \frac{1}{{1\thinspace 024}} \\
    n &= 5 \\
    r &= ? \\
    \end{align}\]
    \[\begin{align}
     t_n &= t_1 r^{n - 1}  \\
     \frac{1}{{1\thinspace 024}} &= t_1 r^{5 - 1}  \\
     \frac{1}{{1\thinspace 024}} &= t_1 r^4  \\
     \frac{1}{{1\thinspace 024r^4 }} &= t_1  \\
     \end{align}\]
    Both equations equal \(t_1 \).
    \[\begin{align}
     \frac{1}{{1\thinspace 024r^4 }} &= \frac{1}{{16r}} \\
     16r &= 1\thinspace 024r^4  \\
     \frac{{16}}{{1\thinspace 024}} &= \frac{{r^4 }}{r} \\
     \frac{1}{{64}} &= r^3  \\
     \sqrt[3]{{\frac{1}{{64}}}} &= r \\
     \frac{1}{4} &= r \\
     \end{align}\]

    Alternatively, you can note that the two known terms are three steps away from each other; therefore,
    \(\begin{array}{l}
     r^3  = \frac{{\frac{1}{{1\thinspace 024}}}}{{\frac{1}{{16}}}} \\
     r^3  = \frac{{16}}{{1\thinspace 024}} \\
     r^3  = \frac{1}{{64}} \\
     \end{array}\)

    The terms in the sequence are:
    \(\frac{1}{4}, {\rm{ }}\frac{1}{{16}}, {\rm{ }}\frac{1}{{64}}, {\rm{ }}\frac{1}{{256}}, {\rm{ }}\frac{1}{{1\thinspace 024}}\)

    The general term is:
    \[\begin{align}
     t_n &= t_1 r^{n - 1}  \\
     t_n &= \frac{1}{4}\left( {\frac{1}{4}} \right)^{n - 1}  \\
     t_n &= \left( {\frac{1}{4}} \right)^{1 + n - 1}  \\
     t_n &= \left( {\frac{1}{4}} \right)^n  \\
     \end{align}\]