8. A geometric series has first term \(a\) and common ratio \(r\).
- Prove that the sum of the first \(n\) terms of this series is given by
$$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$
The second term of a geometric series is - 320 and the fifth term is \(\frac { 512 } { 25 }\)
- Find the value of the common ratio.
- Hence find the sum of the first 13 terms of the series, giving your answer to 2 decimal places.