Answer
$S_5 = 31$
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 $a_1 = 1$ and $n = 5$; however, we need to find $r$. This can be done by setting up a ratio of the second term to the first term:
$r = \frac{2}{1} = 2$
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:
$S_5 = \frac{1(1 - 2^5)}{1 - 2}$
Evaluate exponents first:
$S_5 = \frac{1(1 - 32)}{1 - 2}$
Evaluate what's in parentheses:
$S_5 = \frac{1(-31)}{1 - 2}$
Multiply next:
$S_5 = \frac{-31}{1 - 2}$
Subtract:
$S_5 = \frac{-31}{-1}$
Simplify:
$S_5 = 31$
