Prove that:
1(1!) + 2(2!) + ...... + n(n!) = (n+1)! -1
Here is what I havedone so far but I am stuckk?
(n+1)! -1 +((n+1) (n+1)!)
= (n+1)![1 + (n+1)] -1
I cannot ge it to factor own to what it is suppose to be, please help!
Hello, luckyc1423!
Prove that:
. . 1(1!) + 2(2!) + 3(3!) + ... + n(n!) .= .(n+1)! -1
You should write out the steps . . . clearly.
Verify S(1): .1(1!) .= .2! - 1 . . . true
Assume S(k): .1(1!) + 2(2!) + 3(3!) + ... + k(k!) .= .(k+1)! - 1
We will try to prove that S(k+1) is true.
. . You might write it out (on the side) just for reference.
Add (k+1)(k+1)! to both sides:
. . 1(1!) + 2(2!) + 3(3!) + ... + (k+1)(k+1)! .= .(k+1)! - 1 + (k+1)(k+1)!
The left side is the left side of S(k + 1).
The right side is: .(k+1)! + (k+1)(k+1)! - 1
. . . . . . . Factor: .(k+1)![1 + (k + 1)] - 1
. . And we have: . (k+1)!(k+2) - 1 .= .(k+2)! - 1
This is the right side of S(k+1). .The inductive proof is complete.