N2019 P1 Q8

An arithmetic series has first term \(a\) and common difference \(2a,\)where \(a\ne 0.\) A geometric series has first term \(a\) and common ratio \(2\). The \({{k}^{\text{th}}}\) term of the geometric series is equal to the sum of the first \(64\) terms of the arithmetic series. Find the value of \(k\).[3]

A geometric series has first term \(f\) and common ratio \(r\), where \(f\), \(r\in \mathbb{R}\) and \(f\ne 0\). The sum of the first four terms of the series is \(0\). Find the possible values of \(f\) and \(r.\) Find also, in terms of \(f,\) the possible values of the sum of the first \(n\) terms of the series.[4]

The first term of an arithmetic series is negative. The sum of the first four terms of the series is \(14\) and the product of the first four terms of the series is \(0\). Find the \({{11}^{\mathrm{th}}}\) term of the series.[4]