2014 DHS Promo Q6

A geometric series consisting of positive terms has first term \(a\) and common ratio \(r\).

1. Given that the sum of the first four terms of the series is \(16\) times that of the sum of the next four terms, find the value of \(r\).[3]

An arithmetic series also has first term \(a\). The ratio of its ninth term to the first term is \(r\).

2. Given that the \(k\)th term in the arithmetic series is zero, find \(k\).[2]
3. Find the least value of \(n\) such the sum of the first \(n\) terms of the arithmetic series is less than \(a\).[4]