8. A geometric series is \(a + a r + a r ^ { 2 } + \ldots\)
- Prove that the sum of the first \(n\) terms of this series is \(S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }\).
The first and second terms of a geometric series \(G\) are 10 and 9 respectively.
- Find, to 3 significant figures, the sum of the first twenty terms of \(G\).
- Find the sum to infinity of \(G\).
Another geometric series has its first term equal to its common ratio. The sum to infinity of this series is 10.
- Find the exact value of the common ratio of this series.