题目

Problem

9. (i) A sequence of numbers is defined by

$u_1 = 3$

$u_{n+1} = 2u_n - 2n + 1 \quad n \geq 1$

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

$u_n = 5 \times 2^{n-1} - n \times 2^n$

(5)

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

$f(n) = 5^{n+2} - 4n - 9$

is divisible by $16$

(5)

题目中文翻译

$u_1 = 3$

$u_{n+1} = 2u_n - 2n + 1 \quad n \geq 1$

用数学归纳法证明，对于 $n \in \mathbb{Z}^+$

$u_n = 5 \times 2^{n-1} - n \times 2^n$

(ii) 用数学归纳法证明，对于 $n \in \mathbb{Z}^+$

$f(n) = 5^{n+2} - 4n - 9$

能被 $16$ 整除。

IAL 2022 June FP1 Q9