2026 IGCSE Additional Mathematics May/June 0606/23 Q11

A geometric progression has first term \(a\) and common ratio \(r\), where \(a\ne0\), \(r>0\) and \(r\ne1\).

The terms of the geometric progression are denoted by \(u_1,u_2,u_3,\ldots\).

1. Given that \(u_{n+1}-u_n=\dfrac16(u_{n+3}-u_{n+1})\) show that \(r^2+r-6=0\).[4]
2. It is also given that \(u_{k+2}-u_k=48a\) for some integer \(k\).
   Find the value of \(r\) and the value of \(k\).[4]