This is a example question on a practice exam, any help would be appreciated. It is attached. Using induction to prove it.
Thanks
For all integer $\displaystyle n \geq 1$, $\displaystyle \sum_{i=1}^{n} i \cdot i! = (n+1)! - 1$
Sounds like your typical induction problem. Skipping the formalities ...
Inductive step
Assume it is true for $\displaystyle n = k$, i.e. $\displaystyle \sum_{i=1}^{k} i \cdot i! = (k+1)! - 1$.
It remains to show that it is also true for $\displaystyle n = k+1$, i.e. $\displaystyle \sum_{i=1}^{k+1} i \cdot i! = (k+2)! - 1 \qquad {\color{blue}(1)}$
But:
$\displaystyle \begin{aligned} \sum_{i=1}^{k+1} i \cdot i! & = \overbrace{{\color{red}1 \cdot 1! + 2\cdot 2! + \cdots + k \cdot k!}}^{\displaystyle \text{This is: } \sum_{i=1}^{k} i \cdot i!} + (k+1) \cdot (k+1)! \\ & = {\color{red}(k+1)! - 1} + (k+1)\cdot(k+1)! \qquad {\color{red} \text{By assumption}}\end{aligned}$
Factor out $\displaystyle (k+1)!$ from the expression and you should get it equal to $\displaystyle {\color{blue}(1)}$