Use the Principle of Mathematical Induction to prove that 1

Use the Principle of Mathematical Induction to prove that 1. 1! + 2. 2! + 3. 3! + ... + n. n! = (n + 1)! - 1 for all n greaterthanorequalto 1.

Solution

n = 1, 1*1! = (1+1)!-1 checked.
Assume at n = k, 1.1!+ 2.2!+3.3!...+k.k != (k+1)!-1
At n = k+1,
1.1!+ 2.2!+3.3!...+k*k!+(k+1)(k+1)!
= (k+1)!-1+(k+1)(k+1)!
= (k+1)! (k+2) - 1
= (k+2)! - 1