Prove by induction that = (n+1)!-1
for all integers n>=1
Inductive base case.
Let n=1. Then you have for LHS
(1+1)! - 1 = 2! - 1 = 2-1 = 1. So hold for n=1.
Assume it holds for n, i.e
Try to prove it works for n+1
So for n+1 we want to get .
For the terms involving n take out a common factor of (n+1)! to get...
. which is what we wanted. Hence by our inductive hypothesis the equality holds.