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Summation Notation for a Geometric Series

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Example: Finding the Sum of a Geometric Series Given in Summation Notation

Find the sum displaystylesum_{i=1}^{15} 2(3)^i.

Because the series is expressed in sigma notation, start by identifying the values needed for the sum formula $S_n = \frac{a_1(1 - r^n)}{1 - r}$ : the first term $a_1$ , the common ratio $r$ , and the number of terms $n$ . Write out the first few terms by substituting successive values of $i$ :

$2(3)^1,; 2(3)^2,; 2(3)^3,; \dots$

which gives 6,; 18,; 54,; dots

Identify $a_1$ : the first term is $a_1 = 6$ .

Find the common ratio by dividing consecutive terms: $\frac{18}{6} = 3$ and $\frac{54}{18} = 3$ , so $r = 3$ .

The upper limit of the summation gives $n = 15$ . Substitute $a_1 = 6$ , $r = 3$ , and $n = 15$ into the formula:

$S_{15} = \frac{6(1 - 3^{15})}{1 - 3}$

Simplify:

$S_{15} = 43{,}046{,}718$

This example illustrates how to evaluate a geometric series that is presented in summation notation: expand the first few terms to determine $a_1$ and $r$ , read $n$ from the upper limit, and then apply the partial-sum formula.

Updated 2026-06-03

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OpenStax Intermediate Algebra

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