2014 CJC Promo Q4

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

1. Show that \({{r}^{2}}-r-1=0\).[3]
2. Given that the geometric series is convergent, find the common ratio \(r\), leaving your answer in the exact form.[2]
3. Given that \(a=\frac{1}{r}\) , find the sum of the first 20 terms of the arithmetic series in the form \(p+q\sqrt{5}\), where \(p\) and \(q\) are integers to be determined.[3]