# Factorials, Simplify the addition and multiplication of factorials.

I have an equation containing factorials here

$$(k+1)!+(k+1)(k+1)!$$

yet I am having a hard time understanding how to simplify it using algebra. A simple search on wolfram gets me a reduced form of

$$(k + 2)!$$

This would be a great refresher to such problems, sadly I don't know the elementary operations to reduce it.

• Hint: $a+(k+1)a=(k+2)a$. – dxiv Oct 13 '16 at 3:56

## 2 Answers

\begin{align} &(k + 1)! + (k + 1)(k + 1)!\\ = & (k + 1)!(1 + k + 1)\\ = & (k + 1)!(k + 2)\\ = & (k + 2)(k + 1)(k)(k - 1)....(1)\\ = & (k +2)!\\ \end{align} First you factor out the $(k + 1)!$ and simplify $(1 + k + 1) = (k + 2)$. You are left with $(k + 2)(k + 1)!$ which is just $(k + 2)!$.

• I did not see that. Especially the last three steps, where you an simply remove K + 1 in the third step. – MooCow Oct 13 '16 at 4:41

Factor out the $(k+1)!$:

$$(k+1)! + (k+1)(k+1)! = (1+k+1)(k+1)! = (k+2)(k+1)! = (k+2)!$$

• You are factoring out $(k+2)!$, isn't it? – Sum Mar 25 '17 at 3:36