(a) 使用求和标准公式证明

其中 $a$ 和 $b$ 是待确定的整数。

(b) 用数学归纳法证明，对于 $n \in \mathbb{N}$

使用 (a) 和 (b) 的结果，展示所有运算过程，

(c) 确定满足下式的 $n$ 值

(完全依赖计算器技术的解答不可接受。)

把原和式拆成平方和与常数和，再把平方和公式中的项数取为 $2n$。整理并因式分解后，与题目给出的形式比较，即可确定 $a$、$b$。

Let

Using the standard result

we have

Since

it follows that

Factorising the quadratic,

Therefore,

Comparing this with

gives

仍先使用平方和与常数和公式，但将结果展开成三次多项式，不急着因式分解。把题目指定的含 $a,b$ 的形式也展开，再比较各次项系数。

Let

Using the standard summation formulae,

Expanding the required form,

Comparing the coefficients of $n^3$ and $n$ gives

and

Hence $a=4$ and $b=-1$. These also give $a+b=3$, so the coefficient of $n^2$ is $4$, as required. Thus

严格按归纳法完成三步：先验证 $n=1$；再假设命题在 $n=k$ 时成立，并加上第 $k+1$ 项；最后把结果完全改写成题中公式把 $n$ 换成 $k+1$ 后的形式，再写归纳结论。整理时先提取 $k+1$，再对剩下的二次式分组因式分解。

Let

For $n=1$,

and

Hence the result is true for $n=1$.

Assume that the result is true for $n=k$, so that

Then

To factorise the cubic,

Also,

Therefore,

This is the required formula with $n$ replaced by $k+1$.

Therefore, if the result is true for $n=k$, it is also true for $n=k+1$. Since it is true for $n=1$, by mathematical induction,

for all $n\in\mathbb N$.

把 (a)、(b) 的结果直接代入给定等式。约去非零因子 $n(n+1)$ 后会得到二次方程；因式分解后，还要根据 $n$ 是正整数舍去非整数根。

Using the results from parts (a) and (b),

Since $n$ is a positive integer, $n(n+1)\ne0$, so we may divide by $n(n+1)$. Hence

Expanding and collecting terms,

Thus $n=\frac29$ or $n=5$. Since $n$ must be a positive integer,