# Geometric Series

## Key Questions

• You can find the common ratio $r$ by finding the ratio between any two consecutive terms.

$r = {a}_{1} / {a}_{0} = {a}_{2} / {a}_{1} = {a}_{3} / {a}_{2} = \cdots = {a}_{n + 1} / {a}_{n} = \cdots$

For the geometric series ${\sum}_{n = 0}^{\infty} {\left(- 1\right)}^{n} \frac{{2}^{2 n}}{5} ^ n$, the common ratio

$r = {a}_{1} / {a}_{0} = \frac{- {2}^{2} / 5}{1} = - \frac{4}{5}$

• The sum of a convergent geometric series ${\sum}_{n = 0}^{\infty} a {r}^{n}$ is $\frac{a}{1 - r}$. Since $a = 8$ and $r = \frac{1}{2}$ in the posted geometric series, the sum is $\frac{8}{1 - \frac{1}{2}} = 16$.

• It's the sum of a sequence of terms where each term equals the previous one multiplied by a given ratio (e.g. $1 + 2 + 4 + 8. . .$).