题目

Problem

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

$\sum_{r=1}^{n} \frac{2r^2 - 1}{r^2(r+1)^2} = \frac{n^2}{(n+1)^2}$

(6)

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

$f(n) = 12^n + 2 \times 5^{n-1}$

is divisible by $7$

(6)

题目中文翻译

$\sum_{r=1}^{n} \frac{2r^2 - 1}{r^2(r+1)^2} = \frac{n^2}{(n+1)^2}$

(ii) 用数学归纳法证明，对于 $n \in \mathbb{Z}^+$

$f(n) = 12^n + 2 \times 5^{n-1}$

能被 $7$ 整除。

IAL 2020 May FP1 Q8