Mathematical induction proof

Question

i need to proof the following formula by math. induction

$ \sum_{k=1}^n k!*k=(n+1)!-1 $

n ∈ N

I dont know how to start and there to end, i would appreciate any help.

Answer 1

Basis step : $\sum_{k=1}^{1} k!*k = 1!*1 = 2!-1$

Induction step : $$\begin{align} \sum_{k=1}^n k!*k=(n+1)!-1\\ &\implies \sum_{k=1}^{n+1} k!*k\\ &= \sum_{k=1}^{n} k!*k + (n+1)!*(n+1)\\ &=(n+1)!-1 +(n+1)!*(n+1)\\ &=(n+1)!(n+1+1) -1\\ &=(n+2)! -1 \end{align}$$