Simplifying factorials

Question

I apologise for a repost, but my rep is not high enough to ask in a comment. But, in this question Simplify sum of factorials with mathematical induction I am confused how:

$$(n+1)!-1+(n+1)(n+1)! = -1+(n+1)!(1+(n+1))$$

I expect it's quite simple, but I couldn't for the life of me see the solution. Any help would be much appreciated.

Answer 1

First of all, the actual wording has only one free variable: $$(n+1)! -1+(n+1)\cdot(n+1)! = -1 + (n+1)!\cdot(1+n+1) = (n+2)! - 1$$ All that is used are the usual distributive law, note that $(n+1)!$ is an integer just like any other and the $!$ just binds the immediately preceeding integer, and the equation $$n! = n \cdot (n-1)!$$ Here is the same with some highlighting: $$\begin{align*} \color{red}{(n+1)!} \color{blue}{- 1} + (n+1) \cdot \color{red}{(n+1)!} & \stackrel{y = (n+1)!}= \color{red}y \color{blue}{-1}+(n+1)\cdot \color{red}y\\ & = \color{blue}{-1} + \color{red}y\cdot(1+(n+1))\\ & = \color{blue}{-1} + \color{red}{(n+1)!} \cdot (1 + (n+1)) \\ & = -1 +(n+1)! \cdot (n+2) \\ & = -1 + (n+2)! \end{align*}$$

Answer 2

$n \cdot n! = (n+1-1)n!=(n+1)n!-n!=(n+1)!-n!$
Now, sum of all this taking $n=1,2,3,...$
$1 \cdot 1!+2 \cdot 2! + 3 \cdot 3! + ... + n \cdot n!$$=(2!-1!)+(3!-2!)+(4!-3!)+...+((n+1)!-n!)=(n+1)!-1!$

Answer 3

For example, to simplify $6!$, substitute $6$ for $n$ in the following equation:$$n!=n\cdot(n-1)\cdot(n-2)\cdot(n-3)\cdot...$$$$6!=6\cdot(6-1)\cdot(6-2)\cdot(6-3)\cdot(6-4)\cdot(6-5)$$and now simplify:$$6!=6\cdot5\cdot4\cdot3\cdot2\cdot1$$but you don't have to including the $1$ because it's the same thing as not using it:$$6!=6\cdot5\cdot4\cdot3\cdot2$$Now evaluate it:$$30\cdot4\cdot3\cdot2$$$$120\cdot3\cdot2$$$$360\cdot2$$$$720$$Use the regular equation to help you with not just this, but any whole-number factorial.