Example  2
The sum of a geometric series is \(2\thinspace 046\).  The common ratio is \(4\) and the number of terms is \(5\).  Determine the first term.

Using a formula for the sum of a geometric series, substitute the given values, and solve for \(t_1 \).
\(S_5 = 2\thinspace 046 \)
\(t_1 = ? \)
\(n = 5 \)
\(r = 4 \)
\(\begin{align}
 S_n &= \frac{{t_1 \left( {r^n - 1} \right)}}{{r - 1}} \\
 2\thinspace 046 &= \frac{{t_1 \left( {4^5 - 1} \right)}}{{4 - 1}} \\
 6\thinspace 138 &= t_1 \left( {1\thinspace 023} \right) \\
 6 &= t_1  \\
 \end{align}\)
The first term in the series is \(6\).