2025 RI Promo Q11

An arithmetic series has positive first term \(a\) and common difference \(d\). The first, fifth and second terms of the arithmetic series are the first three consecutive terms of a geometric series respectively. It is also given that the terms of the arithmetic series are decreasing.

1. Show that \(d=-\frac{7}{16}a\).[3]
2. Give a reason why the geometric series converges.[2]

It is given that the sum to infinity of the geometric series is \(\frac{32}{7}\).

4. Find the sum of the first twenty terms of the arithmetic series.[2]
5. Given that the sum of all the terms after the \(n\)th term of the geometric series is greater than 2% of the sum of the first \(n\) terms of the geometric series, find the possible values of \(n\).[4]