# Prove that $n(n^2 - 1) = \frac{(n+1)!}{(n-2)!}$

Prove that for all $n \in \mathbb{N}$,

$$n(n^2 - 1) = \frac{(n+1)!}{(n-2)!}.$$

Thanks in advance.

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Hint: $n(n^2 - 1) = n(n+1)(n-1) = (n+1)(n)(n-1)$ – Isaac Solomon Nov 9 '13 at 16:20
@Pauly As you seem to be new here, I suggest you read this. – Git Gud Nov 9 '13 at 16:21
Do you mean $\frac{(n+1)!}{(n-2)!}$ by any chance? – robjohn Nov 9 '13 at 17:05

## 2 Answers

$\require{cancel}$

If you really want to express $n(n^2 - 1)$ using factorials, note that $$n(n^2 - 1) = n(n+1)(n-1) = (n+1)(n)(n-1) = \frac{(n+1)n(n-1)\cancel{(n-2)}\cdots \cancel{2}\cdot 1}{\cancel{(n - 2)}\cancel{ (n-3)}\cdots \cancel{2}\cdot 1}$$

So we have that $$n(n^2 - 1) = \dfrac{(n+1)!}{(n-2)!}$$

So the given identity you are asked to prove is incorrect: the numerator needs to be $(n + 1)!$.

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Nice guess of the actual question. – Git Gud Nov 9 '13 at 16:34
Thanks, @GitGud I'm developing a knack for that! – amWhy Nov 9 '13 at 16:35
@GitGud I wish the "+" was not the "shift" of "=" (on the same key). I've made the same typo before :-( – amWhy Nov 9 '13 at 16:38
Thank you so much! – Pauly Nov 9 '13 at 16:39
You're welcome, Pauly! – amWhy Nov 9 '13 at 16:39

Here is how

$$n(n-1)=\frac{n(n-1)(n-2)!}{(n-2)!}=\frac{n!}{(n-2)!}$$

$$n+1 = \frac{(n+1)!}{n!}.$$

Can you finish it?

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