How to Simplify Factorials with a Proof for k*(k!)=(k+1)!-1?

[kathrynag]

Homework Statement

I'm trying to prove k*(k!)=(k+1)!-1

Homework Equations

The Attempt at a Solution

This is how far I've gotten:
k[k(k-1)(k-2)...1)]

 

[mutton]

What you are trying to prove is not true for all k.

 

[kathrynag]

Ok, but then what would I do since I know it's true for k=1

 

[mutton]

Are you sure the question doesn't say k * k! = (k + 1)! - k!?

 

[kathrynag]

No, I'm positive. Just checked in the book.

 

[mutton]

Plug in k = 2 to see that it's false.

 

[kathrynag]

Ok, so if I was given this question. I just write only true for k=1?

 

[kathrynag]

Are you sure the question doesn't say k * k! = (k + 1)! - k!?

Well it's 1*1!+2*2!+...+k*k!=(k + 1)! - k!

 

[mutton]

Well it's 1*1!+2*2!+...+k*k!=(k + 1)! - k!

That's not true.

$$k * k! = (k + 1 - 1) k! = (k + 1) k! - 1 * k! = (k + 1)! - k!$$

 

[kathrynag]

Well it's 1*1!+2*2!+...+k*k!=(k + 1)! - k!

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

 

[mutton]

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

Okay, so how do you plan to prove this?

 

[kathrynag]

Never mind. Just figured it out!