Answer
$\dfrac{1024}{5}$
Work Step by Step
An infinite geometric series converges if and only if $|r|\lt1$, and then its sum is equal to $\dfrac{a_1}{1-r}$
where $r$ is known as the common ratio of the quotient of two consecutive terms and $a_1$ is the first term.
Now, the given series has the common ratio:
$r=\dfrac{a_2}{a_1}=\dfrac{-64}{256}=-\dfrac{1}{4}$
Thus, $\left|-\dfrac{1}{4}\right|=\dfrac{1}{4}\lt1$ shows that the series converges.
We are given that $a_1=256$
Thus, the sum is equal to:
$\dfrac{a_1}{1-r} =\dfrac{256}{1-(-\frac{1}{4})}=(256)(\dfrac{4}{5})=\dfrac{1024}{5}$
