# How do you find the common ratio of an infinite geometric series?

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