Example 1
Determine if the following infinite sequences are arithmetic. If the sequence is arithmetic, determine the common difference, \(d \), and give the next three terms in the sequence.

  1. \(1, 3, 5, 7, 9, ... \)

    Determine whether there is a common difference, using pairs of consecutive terms. If there is a common difference, the sequence is arithmetic.

    \(d = t_n - t_{n - 1} \)
    \(d = 3 - 1 = 2 \)
    \(d = 5 - 3 = 2 \)
    \(d = 7 - 5 = 2 \)
    \(d = 9 - 7 = 2 \)

    Yes, this is an arithmetic sequence because each term is two more than the one previous; therefore, the common difference, \(d \), is \(2\).

    The next three terms are:

    \(t_6 = 9 + 2 = 11 \)
    \(t_7 = 11 + 2 = 13 \)
    \(t_8 = 13 + 2 = 15 \)

  2. \(199, 145, 91, 37, -17, ... \)

    Compare the differences of each pair of consecutive terms.

    \( \begin{array}{l}
    d = t_n - t_{n - 1} \\
    d = 145 - 199 = -54 \\
    d = 91 - 145 = -54 \\
    d = 37 - 91 = -54 \\
    d = -17 - 37 = -54 \\
    \end{array}\)

    Yes, this is an arithmetic sequence because each term differs from the one previous by \(-54\) ; therefore, the common difference, \(d \) , is \(-54\).

    The next three terms are:

    \(t_6 = -17 - 54 = -71 \)
    \(t_7 = -71 - 54 = -125 \)
    \(t_8 = -125 - 54 = -179 \)

  3. \(8, 13, 19, 26, 34 \)

    Compare the differences of each pair of consecutive terms.

    \(\begin{array}{l}
    d = t_n - t_{n - 1} \\
    d = 13 - 8 = 5 \\
    d = 19 - 13 = 6 \\
    \end{array}\)

    While you could continue to check all the other pairs of consecutive terms, because the first two pairs do not have a common difference, this is not an arithmetic sequence. The difference must be constant for pairs of consecutive terms in order for the sequence to be arithmetic.