Lesson 4: Geometric Series

Property

To derive the geometric series formula, multiply the series by the common ratio $r$ and subtract from the original series to eliminate most terms: If $S = a + ar + ar^2 + \cdots + ar^{n-1}$, then $rS = ar + ar^2 + ar^3 + \cdots + ar^n$, so $S - rS = a - ar^n$.

Examples

Property

When the common ratio $r = 1$ in a geometric series, all terms are equal to the first term $a$. The sum of $n$ terms is:

Property

The sum, $S_n$, of the first $n$ terms of a geometric sequence is

where $a_1$ is the first term and $r$ is the common ratio, and $r$ is not equal to one.