Sum to infinity from S_n ratio

10. In a geometric series the common ratio is \(r\) and sum to \(n\) terms is \(S _ { n }\) Given $$S _ { \infty } = \frac { 8 } { 7 } \times S _ { 6 }$$ show that \(r = \pm \frac { 1 } { \sqrt { k } }\), where \(k\) is an integer to be found.

8. A geometric series has first term \(a\) and common ratio \(r\) where \(r > 1\). The sum of the first \(n\) terms of the series is denoted by \(S _ { n }\). Given that \(S _ { 4 } = 10 \times S _ { 2 }\),

1. find the value of \(r\). Given also that \(S _ { 3 } = 26\),
2. find the value of \(a\),
3. show that \(S _ { 6 } = 728\).

The first term of a convergent geometric progression is 10. The sum of the first 4 terms of the progression is \(p\) and the sum of the first 8 terms of the progression is \(q\). It is given that \(\frac{q}{p} = \frac{17}{16}\). Find the two possible values of the sum to infinity. [5]