Factorial simplication question

Question

How does the following:

$$(k+1)! - 1 + (k+1).(k+1)!$$ simplify to:

$$ (1+k+1).(k+1)! - 1 $$

and then

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

I just can't seem to see how that works, I've tried writing out the factorials in the form:

$ (k+1)(k)(k-1)....$ and looking at in this way but it still isn't clicking.

Any help would be great.

Thanks!

Answer 1

Factor out $(k+1)!$ from the first equation. Then it should easily follow.

Answer 2

\begin{align} (k+1)! - 1 + (k+1)(k+1)! &= 1\cdot(k+1)! + (k+1)\cdot(k+1)! \\ &=\big((k+1)+1\big)\cdot(k+1)!-1 \\ &= (k+2)(k+1)! - 1 \\ &= (k+2)! - 1. \end{align}