Example 4
Given that the following sequences are geometric, determine the missing terms.

  1. \(\_\_\_, \_\_\_, 6, 36, \_\_\_\)

    Because the question states that the sequence is geometric, and you are given two consecutive terms, you can determine \(r \).
    \(r = \frac{t_n}{t_{n - 1}} = \frac{36}{6} = 6 \)

    Then, to find the missing terms, divide by \(r \) to go down in term number, and multiply by \(r \) to go up in term number.
    \[\begin{align}
     t_2 &= 6 \div 6 = 1 \\
     t_1 &= 1 \div 6 = \frac{1}{6} \\
     t_5 &= 36\left( 6 \right) = 216 \\
     \end{align}\]

    The sequence is \(\frac{1}{6}, 1, 6, 36, 216 \).
  2. \(768, \_\_\_, \_\_\_, 96, \_\_\_\)

    Because this question does not give two consecutive terms, it is harder to determine \(r \). One method is to guess and check, but that is not very efficient! Instead, substitute the known values into the general term. You are given \(t_1 = 768 \), \(t_4 = 96 \), and \(n = 4 \), so substitute these values into the general term formula, and solve for \(r \).

    \[\begin{align}
     t_n &= t_1 \left( r \right)^{n - 1}  \\
     96 &= 768\left( r \right)^{4 - 1}  \\
     \frac{1}{8} &= r^3  \\
     \sqrt[3]{{\frac{1}{8}}} &= \sqrt[3]{{r^3 }} \\
     \frac{1}{2} &= r \\
     \end{align} \]

    Therefore, the missing terms are:
    \[\begin{align}
     t_2 &= 768\left( {\frac{1}{2}} \right) = 384 \\
     t_3 &= 384\left( {\frac{1}{2}} \right) = 192 \\
     t_5 &= 96\left( {\frac{1}{2}} \right) = 48 \\
     \end{align} \]

    The sequence is \(768, 384, 192, 96, 48\).

    Recall that when you take the \(x \) root of a variable with an exponent of \(x \), you are just left with the variable (and an exponent of 1).
    \(\sqrt[x]{{y^x }} = y \)

    If you need a review on how to evaluate radicals using your calculator, please view the Calculator Guide.

    Also note that if given the option to use decimals or fractions, please use fractions. The further you go in mathematics, the more fractions are preferred over decimals.


  3. \(\_\_\_, \_\_\_, 896, \_\_\_, 14\thinspace 336\)

    This is the most difficult situation. You do not know \(t_1 \) or \(r \). But, you do know two terms, and you know that the value of \(t_1 \) and \(r  \) are the same for both. You will need to solve a system of equations to solve for the two unknowns.

    Set up two equations, and isolate \(t_1 \) in each.

    Equation I
    \(\begin{align}
    t_1 &= ? \\
    r &= ? \\
    t_3 &= 896 \\
    n &= 3 \\
    \end{align}\)
    \[\begin{align}
     t_n &= t_1 r^{n - 1}  \\
     896 &= t_1 r^{3 - 1}  \\
     896 &= t_1 r^2  \\
     \frac{{896}}{{r^2 }} &= t_1  \\
     \end{align} \]
    Equation II
    \(\begin{align}
    t_1 &= ? \\
    r &= ? \\
    t_5 &= 14\thinspace 336 \\
    n &= 5 \\
    \end{align}\)
    \[\begin{align}
     t_n &= t_1 r^{n - 1}  \\
     14\thinspace 336 &= t_1 r^{5 - 1}  \\
     14\thinspace 336 &= t_1 r^4  \\
     \frac{{14\thinspace 336}}{{r^4 }} &= t_1  \\
     \end{align} \]
    Because both equations equal \(t_1 \), you can equate the two equations and solve for \(r \).

    \[\begin{align}
     \frac{{896}}{{r^2 }} &= \frac{{14\thinspace 336}}{{r^4 }} \\
     896r^4 &= 14\thinspace 336r^2  \\
     \frac{{r^4 }}{{r^2 }} &= \frac{{14\thinspace 336}}{{896}} \\
     r^{4 - 2} &= 16 \\
     r^2 &= 16 \\
     r &= \pm \sqrt {16}  \\
     r &= \pm 4 \\
     \end{align} \]

    You may be able to look at the sequence and notice that \(t_3 \) and \(t_5 \) are two steps away from each other; therefore, the ratio of those two terms is equal to \(r^2 \).

    You could then jump to \(r^2 = \frac{14\thinspace 336}{896} \), and solve for \(r \).

    Note that when you take the square root of a squared variable, it is possible that the value of the variable is negative or positive. Both \((–4)^2\) and \((4)^2\) give \(16\);  therefore, \(r \) can be both \(–4 \) and \(4\).

    Using these values of \(r \), you can work backwards from the known terms to find the missing terms. There will be two sequences that are possible.
    \(\begin{array}{l}
     t_2 = 896 \div 4 = 224 \\
     t_1 = 224 \div 4 = 56 \\
     t_4 = 14\thinspace 336 \div 4 = 3\thinspace 584 \\
     \end{array} \)
    \(\begin{array}{l}
     t_2 = 896 \div - 4 = -224 \\
     t_1 = -224 \div - 4 = 56 \\
     t_4 = 14\thinspace 336 \div -4 = -3\thinspace 584 \\
     \end{array}\)
    The sequences are \(56, 224, 896, 3\thinspace 584, 14\thinspace 336\) or \(56, -224, 896, -3\thinspace 584, 14\thinspace 336\).