Finite and Infinite Geometric Sums

Table of Contents

Finite Geometric Series #

For a series of the form $ \small a, ar, ar^2, ar^3, \ldots$, the sum of the first $\small n$ terms is

$$S_n = \sum_{i=0}^{n-1} ar^i = \frac{a(1-r^n)}{1-r}$$

where $\small r >0, \neq 1 $ is the common ratio. Note that this expression can be used for $\small r > 1$ unlike the infinite series formula below.

If $\small r = 1$, then $\small S_n = an$.

If $\small r$ satisfies $\small 0 < r < 1$, the sum of the infinite series is

$$S_{\infty} = \sum_{i=0}^{\infty} ar^i = \frac{a}{1-r}$$