Answer
$S_4 = \frac{259}{18}$
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 $a_1 = 12$ and $n = 4$; however, we need to find $r$. This can be done by setting up a ratio of the second term to the first term:
$r = \frac{2}{12} = \frac{1}{6}$
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:
$S_4 = \frac{12(1 - (\frac{1}{6})^4)}{1 - \frac{1}{6}}$
Evaluate exponents first:
$S_4 = \frac{12(1 - \frac{1}{1296})}{1 - \frac{1}{6}}$
Evaluate what's in parentheses:
$S_4 = \frac{12(\frac{1295}{1296})}{1 - \frac{1}{6}}$
Subtract:
$S_4 = \frac{12(\frac{1295}{1296})}{\frac{5}{6}}$
Multiply next:
$S_4 = \frac{\frac{15,540}{1296}}{\frac{5}{6}}$
Divide fractions by multiplying the numerator by the reciprocal of the denominator:
$S_4 = \frac{15,540}{1296} \times \frac{6}{5}$
Multiply:
$S_4 = \frac{93,240}{6480}$
Simplify by dividing both numerator and denominator by their greatest common factor, which is $360$:
$S_4 = \frac{259}{18}$
