Induction with Factorials

Hi, can anyone help me with this induction question

(1x1!)+(2x2!)+(3x3!)+...+(nxn!)

a) Write a conjecture
b) Prove it by induction

a) the conjecture is S: (n+1)! -1
but i dont know how to get this.
thanks!

Reply 1

Chewwy

is it true for n =1? yes.

suppose S_k = (k+1)! -1
S_(k+1) = ((k+1)! - 1) + (k+1)(k+1)! = (k+1)! - 1 + k(k+1)! + (k+1)! = (k+2)(k+1)! - 1 = (k+2)! - 1

Reply 2

nota bene

it is true for n=1 (show it) assume true for n=k, then for the inductive step we add the next term to the RHS and start working...

(k+1)!-1+(k+1)(k+1)!

(k+1)!(1+(k+1)) -1

Now we are nearly there... can you see how to proceed?

Reply 3

t47

i still dont understand how you get Sum: (n+1)!-1
I only got (n+1)!-1 from the answers

Reply 4

ChiefNX

t47

i still dont understand how you get Sum: (n+1)!-1
I only got (n+1)!-1 from the answers

Hmmm? I don't follow you.

Anway, the big trick here is to remember that (n+1)! = (n+1)n! which is the fundamental property of factorials.

Take another look at choochie's solution and try embroidering it with your normal induction routine. Hopefully it will become clear.