Compare Your Answers
  1. Calculate the sum of the following arithmetic series:

    \(18 + 5 - 8 - 21 - ... - 164 \)

    Regardless of which formula you want to use, both need the value of \(n \), so start by finding \(n \) using the general term formula.
    \(\begin{align}
     t_1 &= 18 \\
     t_n &= -164 \\
     d &= 5 - 18 = -13 \\
     n &= ? \\
     \end{align}\)
    \(\begin{align}
     t_n &= t_1 + \left( {n - 1} \right)d \\
      -164 &= 18 + \left( {n - 1} \right)\left( {-13} \right) \\
      -182 &= -13n + 13 \\
      -195 &= -13n \\
     15 &= n \\
     \end{align}\)
    Then, use the value of \(n \) in either of the sum of an arithmetic series formulas.

    \(\begin{align}
     S_n &= \frac{n}{2}\left[ {t_1 + t_n } \right] \\
     S_{15} &= \frac{{15}}{2}\left[ {18 - 164} \right] \\
     S_{15} &= 7.5\left( {-146} \right) \\
     S_{15} &= -1095 \\
     \end{align}\)
    The sum of the series is \(-1095\).

  2. Calculate the sum of the first \(25\) terms of the following arithmetic series:

    \[\frac{1}{4} + \frac{1}{2} + \frac{3}{4} + ...\]

    This time, you are given the value of \(n \), but you do not have the value of \(t_n \), so the longer version of the sum of an arithmetic series formula can be used.
    \[\begin{align}
    t_1 &= \frac{1}{4} \\
    n &= 25 \\
    d &= \frac{1}{4} \\
    S_n &= ?
    \end{align}\]
    \[\begin{align}
     S_n &= \frac{n}{2}\left[ {2t_1 + (n - 1)d} \right] \\
     S_{25} &= \frac{{25}}{2}\left[ {2\left( {\frac{1}{4}} \right) + \left( {25 - 1} \right)\left( {\frac{1}{4}} \right)} \right] \\
     S_{25} &= 12.5\left[ {\frac{1}{2} + 6} \right] \\
     S_{25} &= 81.25 \\
     \end{align}\]

    The sum of the first \(25\) terms is \(81.25\).

  3. Find \(t_{15} \) and \(S_{15} \) of the following series:

    \(2.4 + 2.8 + 3.2 + ... \)

    Use the general term for arithmetic sequences to find \(t_{15} \).
    \[\begin{align}
    t_1 &= 2.4 \\
    t_{15} &= ? \\
    n &= 15 \\
    d &= 2.8 - 2.4 = 0.4 \\
    \end{align}\]
    \(\begin{align}
     t_n &= t_1  + \left( {n - 1} \right)d \\
     t_{15} &= 2.4 + \left( {15 - 1} \right)0.4 \\
     t_{15} &= 2.4 + 5.6 \\
     t_{15} &= 8 \\
     \end{align} \)
    Then, use either arithmetic series formula to find \(S_{15} \).
    \[
    \begin{align}
     S_n &= \frac{n}{2}\left[ {t_1 + t_n } \right] \\
     S_{15} &= \frac{{15}}{2}\left[ {2.4 + 8} \right] \\
     S_{15} &= 7.5\left( {10.4} \right) \\
     S_{15} &= 78 \\
     \end{align}
    \]