<s>[INST] Prove by Induction: Summation of Factorial (n! * n) Prove by induction (weak or strong) that:  
$$(1! \cdot 1) + (2! \cdot 2) + \cdots + (n! \cdot n) =\sum_{k=1}^nk!\cdot k= (n + 1)! - 1$$
My base case is:  
$n = 1$, which is true.
And my Inductive Hypothesis is:  
$(1! \cdot 1) + (2! \cdot 2) + \cdots + (k! \cdot k) = (k + 1)! - 1$
After that, I'm trying to show the $(k + 1)$-stage where:  
$(1! \cdot 1) + (2! \cdot 2) + \cdots + (k! \cdot k) + ((k + 1)! \cdot (k + 1)) = ((k + 1) + 1)! - 1$  
Which simplifies to:
$(1! \cdot 1) + (2! \cdot 2) + \cdots + (k! \cdot k) + ((k + 1)! \cdot (k + 1)) = (k + 2)! - 1$
I see that I can substitute in my Inductive Hypothesis but where I'm stuck is manipulating the LHS to be equal to the RHS after that:  
$(k + 1)! - 1 + ((k + 1)! \cdot (k + 1)) = (k + 2)! - 1$
 [/INST] \begin{align*}
(1!\cdot 1) + (2!\cdot 2) + \cdots + (k!\cdot k) + ((k+1)!\cdot(k+1))
  &= (k+1)!-1 + ((k+1)!\cdot(k+1)) \\
  &= ((k+1)!\cdot(k+2))-1 \\
  &= (k+1)!-1.
\end{align*}
</s>