4.07: Geometric progression differences

0606/23/M/J/26 — Question 11 · 8 marks

A geometric progression has first term \(a\ne0\), common ratio \(r>0\), \(r\ne1\), and terms \(u_n\).
(a)
Given \(u_{n+1}-u_n=\dfrac16(u_{n+3}-u_{n+1})\), show that \(r^2+r-6=0\).
[4]
(b)
Given \(u_{k+2}-u_k=48a\), where \(k\) is an integer, find \(r\) and \(k\).
[4]