Problem

The geometric progression $a_1, a_2, a_3, \ldots$ has first term $2$ and common ratio $r$ where $r > 0$. It is given that $\frac{9}{2}a_5 + 7a_3 = 8$.

(a) Find the value of $r$.

(b) Find the sum of the first $20$ terms of the geometric progression. Give your answer correct to $4$ significant figures.

(c) Find the sum to infinity of the progression $a_2, a_5, a_8, \ldots$.