Theorem For any integer n 1 Proof by mathematical induction

Theorem: For any integer n ? 1,

“Proof (by mathematical induction): Let the property

Show that P(1) is true: When n = 1

So 1(1!) = 2! ? 1
and 1 = 1
Thus P(1) is true.”

the statement is true for n=1 (as proved in the question)

let the statement be true for n = k

1*1! + 2*2! + ..+k*k! = (k+1)!-1

1*1! + 2*2! + ..+k*k! + (k+1)*(k+1)! = (k+1)!-1 + (k+1)*(k+1)!

1*1! + 2*2! + ..+k*k! + (k+1)*(k+1)! = [1+k+1](k+1)!-1

1*1! + 2*2! + ..+k*k! + (k+1)*(k+1)! = [k+2](k+1)!-1 = (k+2)! -1

the statement is true for n = k+1

thus proved by induction