Example 3

The sum of the first nine terms in an arithmetic series is \(459\). The sum of the first ten terms in the same arithmetic series is \(555\).

1. Determine the value of \(t_{10}\).
   
   
   Solution

   You are given two consecutive sums; therefore, you can calculate the last term added, \(t_{10}\).
   

   \(\begin{align}
   S_{10} &= 555 \\
   S_9 &= 459 \\
   t_{10} &= ? \\
   \end{align}\)
   

   \(\begin{align}
    S_n - S_{n - 1} &= t_n  \\ 
   S_{10} - S_9 &= t_{10}  \\
    555 - 459 &= t_{10}  \\
    96 &= t_{10}  \\
    \end{align} \)
   
   
   
2. Determine the value of \(t_1\).
   
   
   Solution

   Using the sum of the first ten terms and the value of the tenth term, you can work backwards to determine the first term.

   \(\begin{align}
   t_1 &= ? \\
   t_{10} &= 96 \\
   S_{10} &= 555 \\
   n &= 10 \
   \end{align}\)
   

   \[\begin{align}
    S_n &= \frac{n}{2}\left( {t_1 + t_n } \right) \\
    555 &= \frac{{10}}{2}\left( {t_1 + 96} \right) \\
    111 &= t_1 + 96 \\
    15 &= t_1  \\
    \end{align} \]

For further information about arithmetic series, please read through pp. 22 - 25 in Pre-Calculus 11.