Example  1
\(6, 11, 16, 21, ..., 66\) is an arithmetic sequence.

  1. Write the sequence as an arithmetic series.

    Written as a series, the sequence would be \(6 + 11 + 16 + 21 + ... + 66\).

  2. Determine the number of terms in the series.

    In Lesson 1.1, you found the number of terms, \(n \), in an arithmetic sequence using the general term formula and the last term in the sequence.

    \(\begin{align}
     t_1 &= 6 \\
     t_n &= 66 \\
     d &= 11 - 6 = 5 \\
     n &= ? \\
     \end{align}\)
    \(\begin{align}
     t_n &= t_1 + \left( {n - 1} \right)d \\
     66 &= 6 + \left( {n - 1} \right)5 \\
     60 &= 5n - 5 \\
     65 &= 5n \\
     13 &= n \\
     \end{align}\)
    There are \(13\) terms in the series.

  3. Calculate the sum of the arithmetic series.

    Either formula will work for this question. The second version is the most simple.

     

    \(\begin{align}
     t_1 &= 6 \\
     t_n &= 66 \\
     n &= 13 \\
     \end{align}\)
    \(\begin{align}
     S_n &= \frac{n}{2}\left[ {t_1 + t_n } \right] \\
     S_{13} &= \frac{{13}}{2}\left[ {6 + 66} \right] \\
     S_{13} &= 6.5\left( {72} \right) \\
     S_{13} &= 468 \\
     \end{align}\)
    The sum of the series is \(468\).