Question

Mathematics Inclined Week #1: Difficult Category

Problem:
Prove, by mathematical induction, that:


`1(1!) + 2(2!) + 3(3!) + ... + n(n!) = (n + 1)! - 1`

You must show your working out either as a screenshot or by LaTeX.
Full solutions will be posted next week, along with a new question.

Answer 1

Let
`n=1`. Then the statement says


`1(1!)=1`

`(1+1)!-1=2!-1=2-1=1`

so it holds for the base case.

Assume it holds for
`n=k`, i.e. that


`1(1!)+2(2!)+3(3!)+\cdots+k(k!)=(k+1)!-1`

and use this to show it holds for
`n=k+1`. You have


`1(1!)+2(2!)+\cdots+k(k!)+(k+1)(k+1)!=(k+1)!-1+(k+1)(k+1)!`

`=(k+1)!(1+k+1)-1`

`=(k+1)!(k+2)-1`

`=(k+2)!-1`

which is what you needed to show, so the statement is true.