N2005 P2 Q4

It is given that \(a\), \(b\), \(c\) are the first three terms of a geometric progression. It is also given that \(a\), \(c\), \(b\) are the first three terms of an arithmetic progression.

1. Show that \({{b}^{2}}=ac\) and \(c=\frac{a+b}{2}\).[2]
2. Hence show that \(2{{\left( \frac{b}{a} \right)}^{2}}-\left( \frac{b}{a} \right)-1=0\).[2]
3. Given that the sum to infinity of the geometric progression is \(S\), find \(S\) in terms of \(a\).[4]