N2007 P1 Q10

A geometric series has common ratio \(r\), and an arithmetic series has first term \(a\) and common difference \(d\), where \(a\) and \(d\) are non-zero. The first three terms of the geometric series are equal to the first, fourth and sixth terms respectively of the arithmetic series.

1. Show that \(3r^2-5r+2=0\).[4]
2. Deduce that the geometric series is convergent and find, in terms of \(a\), the sum to infinity.[5]
3. The sum of the first \(n\) terms of the arithmetic series is denoted by \(S\). Given that \(a>0\), find the set of possible values of \(n\) for which \(S\) exceeds \(4a\).[5]