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FM June 2025 p12 q03
4109
The sequence \(u_1, u_2, u_3, \ldots\) is such that \(u_1 = 5\) and \(u_{n+1} = 6u_n + 5\) for \(n \geq 1\).
(a) Prove by induction that \(u_n = 6^n - 1\) for all positive integers \(n\).
(b) Deduce that \(u_{2n}\) is divisible by \(u_n\) for \(n \geq 1\).
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