题目

Problem

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

$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n + 1)^2$

(5)

(b) Using the standard summation formulae, show that

$\sum_{r=1}^{n} r(r + 1)(r - 1) = \frac{1}{4}n(n + A)(n + B)(n + C)$

where $A$ , $B$ and $C$ are constants to be determined.

(4)

$3\sum_{r=1}^{n} r(r + 1)(r - 1) = 17\sum_{r=n}^{2n} r^2$

(5)

题目中文翻译

(a) 用数学归纳法证明：对于 $n \in \mathbb{Z}^+$ ， $\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n + 1)^2$

(b) 使用标准求和公式证明 $\sum_{r=1}^{n} r(r + 1)(r - 1) = \frac{1}{4}n(n + A)(n + B)(n + C)$ 其中 $A$ 、 $B$ 、 $C$ 为待确定的常数。

IAL 2022 Jan FP1 Q9