Answer
$S_8 = 53.125$
Work Step by Step
To find the sum of a finite geometric series, we can use the formula:
$S_n = \frac{a_1(1 - r^n)}{1 - r}$, $r \ne 1$, where $S_n$ is the sum of the series, $n$ is the number of terms in the series, $a_1$ is the first term, and $r$ is the common ratio.
We already have $a_1 = 80$ and $n = 8$; however, we need to find $r$. This can be done by setting up a ratio of the second term to the first term:
$r = \frac{-40}{80} = -\frac{1}{2}$
Now that we have the common ratio, we can now plug in our values into the formula to calculate the sum of a geometric series:
$S_8 = \frac{80(1 - (-\frac{1}{2})^8)}{1 - (-\frac{1}{2})}$
Evaluate exponents first:
$S_8 = \frac{80(1 - \frac{1}{256})}{1 - (-\frac{1}{2})}$
Evaluate what's in parentheses:
$S_8 = \frac{80(\frac{255}{256})}{1 + \frac{1}{2}}$
Add:
$S_8 = \frac{80(\frac{255}{256})}{\frac{3}{2}}$
Multiply next:
$S_8 = \frac{\frac{20,400}{256}}{\frac{3}{2}}$
Divide fractions by multiplying the numerator by the reciprocal of the denominator:
$S_8 = \frac{20,400}{256} \times \frac{2}{3}$
Multiply:
$S_8 = \frac{40,800}{768}$
Simplify:
$S_8 = 53.125$
