Problem

The first three terms of a geometric progression are $a$, $b$ and $c$ respectively, where $a$, $b$ and $c$ are positive constants. The first three terms of an arithmetic progression are $a$, $b$ and $-3c$ respectively.

(a) Show that $a^2-10ac+9c^2=0$.

It is now given that $a=9$ and $c$ takes the smaller of its two possible values.

(b) (i) Find the sum to infinity of the geometric progression.

(ii) Find the sum of the first 20 terms of the arithmetic progression.