2013 NJC P1 Q4

A convergent geometric progression has a positive first term \(a\) and common ratio \(r\) while an arithmetic progression has a non-zero common difference \(d\). The second and third term of the geometric progression is the first and fourth term of the arithmetic progression respectively. It is also given that the first term of the geometric progression is equal to the sum of the first \(4\) odd-numbered terms in the arithmetic progression.

1. By expressing the first term of the arithmetic progression in terms of \(a\) and \(r\), show that \(r^2 = \frac{1}{4}\).[4]
2. It is given that the sum to infinity of the geometric progression is less than \(a\).
   1. Find the value of \(r\), justifying your answer.[2]
   2. Explain if this implies that for all \(n\), the sum of the first \(n\) terms of the geometric progression must be less than \(a\).[1]