Help proof by induction I just got my first induction assignment in a new course. They want me to prove by induction that:
$$\sum_{s=1}^k s*s! = (K+1)!-1.$$
The way I understand induction is that I test for the first value. Then for n and then n+1, to show the given expression is true.
I've done it for 1 and n, however I'm stuck at n+1.
$$\sum_{s=1}^{n+1} s*s!=(n+1)!-1+(n+1)*(n+1)!$$
Using maple I can see the expression is $-1+(n+2)!$ (which is true) however i dont know how to reduce/rewrite $(n+1)!-1+(n+1)(n+1)!$ to $-1+(n+2)!.$$
I've asked my friends and a older student, but to no avail. I'm hoping you guys can help.

First proof the statement for some value, which you did (induction basis).
Then induction step: Let's assume that the statement you want to proof holds for K = n. Then you have to establish that it also holds for n+1.
So we know that $\sum_{s=1}^{n} s \cdot s! = (n+1)! - 1$ (induction hypothesis)
$\sum_{s=1}^{n+1} s \cdot s! = \sum_{s=1}^{n} s \cdot s! + (n+1) \cdot (n+1)! = (n+1)! - 1 + (n+1) \cdot (n+1)! = (n+1)! + (n+1) \cdot (n+1)! - 1 = (1 + (n+1)) \cdot (n+1)! - 1 = (n+2) \cdot (n+1)! - 1 = (n+2)! - 1 = ((n+1)+1)! - 1$
So then we know that the statement also holds for n+1. This concludes the proof.
So in proofs by induction you explicitly use the induction hypothesis for the induction step.
