2024 TMJC P1 Q5

An arithmetic progression \(A\) has first term \(a\) and common difference \(d\), where \(a\) and \(d\) are non-zero. A geometric progression \(G\) has first term \(b\) and common ratio \(r\).

1. The first, third and eleventh terms of \(A\) are equal to the fourth, third and second term of \(G\) respectively. Prove that the geometric series of \(G\) is convergent.[4]
2. It is given instead that \(r=\frac{1}{3}\) and the terms of another sequence \(H\) is formed by squaring the terms of \(G\). Find the range of values of \(b\) such that the sum to infinity of \(H\) exceeds the sum to infinity of \(G\) by more than \(\frac{3}{2}\).[3]