2020 RI P2 Q2

In a geometric series, the ratio of the sum of the first \(8\) terms to the sum of the first \(4\) terms is \({17:16}\).
Find the \(2\) possible values of the common ratio, \({{r}_{1}}\), \({{r}_{2}}\) where \({{r}_{1}}>{{r}_{2}}\).The sum to infinity of the geometric series with first term \(a\) and common ratio \({{r}_{1}}\) is denoted by \({{S}_{1}}\) and the sum to infinity of the geometric series with first term \(b\) and common ratio \({{r}_{2}}\) is denoted by \({{S}_{2}}\) . Find \({{S}_{1}}:{{S}_{2}}\) in terms of \(a\) and \(b\).[5]