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Sum of Finite Geometric Series

For finding the sum of terms in a finite geometric series, we use the following formula where $a_{1}$ = the first term in the series and $r$ = the common ratio

(1) $\begin{equation*} S_{n} = a_{1} * (\frac{1-r^{n}}{1-r})\end{equation*}$

(2) $\begin{equation*} S_{n} = \frac{a_{1} - a_{1} * r^{n}}{1-r}\end{equation*}$

Example 1:

Find the sum of the first 6 terms of the geometric series 2 + 4 + 8 + …

We’ll use the first equation knowing that $n = 6$ , $a_{1} = 2$ , and $r = \frac{4}{2} = 2$

(3) $\begin{equation*} S_{6} = 2 * (\frac{1 - 2^{6}}{1 - 2}) = 2 * (\frac{1-64}{1-2}) = 2 * (\frac{-63}{-1}) = \boxed{126}\end{equation*}$

Example 2:

Find the sum of the first n terms of the geometric series given $a_{1} = 2$ , $a_n = 600$ , and $r = 2$

(4) $\begin{equation*} S_{n} = \frac{2 - (600*2)}{1 - 2} = \frac{2 - 1200}{1-2} = \frac{-1198}{-1} = \boxed{1198}\end{equation*}$