<s>[INST] Prove by induction: $1(1!)+\cdots + n\cdot n!$ = (n+1)! - 1 Induction step.
$1(1!) + ... + n(n!) = (n+1)!  - 1$ 
$1(1!) + ... + n(n!) + (n+1)(n+1)! = (n+1)! - 1 + (n+1)(n+1)!$ 
So, I don't understand how to get $(n+2)! - 1$ from $(n+1)! - 1 + (n+1)(n+1)!$.
Please, explain it step by step.
 [/INST] Do some factoring:
\begin{align*}
(n+1)!−1+(n+1)(n+1)!
&= [(n+1)(n+1)! + 1(n+1)!] - 1 \\
&= [(n+1) + 1](n + 1)! - 1 \\
&= (n + 2)(n + 1)! - 1 \\
&= (n + 2)! - 1 \\
\end{align*}
</s>