IAL 2026 Jan FP1 Q10 | 國際數學站

Question

[!problem]

A sequence of numbers $u_1, u_2, u_3, \ldots$ is defined by

$u_1 = 8, \quad u_2 = 10$ $u_{n+2} = u_{n+1} + 2u_n, \quad n \geq 1$

Prove by induction that, for $n \in \mathbb{Z}^+$

$u_n + 3 = 2^{n+2} + (-1)^n$

(6)

中文翻译

一个数列 $u_1, u_2, u_3, \ldots$ 由下式定义

$u_1 = 8, \quad u_2 = 10$ $u_{n+2} = u_{n+1} + 2u_n, \quad n \geq 1$

用数学归纳法证明，对于 $n \in \mathbb{Z}^+$

$u_n + 3 = 2^{n+2} + (-1)^n$

(6)