A geometric series has first term $a = 360$ and common ratio $r = \frac{7}{8}$ - Edexcel - A-Level Maths Pure - Question 3 - 2012 - Paper 4

To find the sum of the first $n$ terms of a geometric series, we use the formula:
$S_n = S = \frac{a(1 - r^n)}{1 - r}$
for $n = 20$.
Substituting the known values:
$S_{20} = \frac{360(1 - (\frac{7}{8})^{20})}{1 - \frac{7}{8}}$
Calculating this gives:
$S_{20} \approx 2680$
Thus, the sum of the first 20 terms of the series is approximately 2680.

For a geometric series, when the absolute value of the common ratio $|r| < 1$, the sum to infinity is given by:
$S_{\infty} = \frac{a}{1 - r}$
Using the values:
$S_{\infty} = \frac{360}{1 - \frac{7}{8}}$
Calculating this gives:
$S_{\infty} \approx 2880$
Thus, the sum to infinity of the series is approximately 2880.