Answer
$S = 187.5$
Work Step by Step
To determine whether or not this series converges or diverges, we need to find the common ratio, $r$. This can be done by setting up a ratio of the second term to the first term:
$r = \frac{30}{150} = \frac{1}{5}$
When $|r| \lt 1$, then the series converges; when $|r| \gt 1$, then the series diverges. Since $|r| \lt 1$ in this case, this series converges.
To find the sum of this convergent 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 $a_1 = 150$, $n = \frac{1}{5}$, and $r = \frac{1}{5}$.
We can now use the following formula to calculate the sum of a convergent geometric series:
$S = \frac{a_1}{1 - r}$
Plug in our known values:
$S = \frac{150}{1 - \frac{1}{5}}$
Subtract first:
$S = \frac{150}{\frac{4}{5}}$
Divide fractions by multiplying the numerator by the reciprocal of the denominator:
$S = \frac{150}{1} \times \frac{5}{4}$
Multiply:
$S = \frac{750}{4}$
Simplify:
$S = 187.5$
