# Need help with a proof using factorials

• Oct 11th 2009, 05:12 PM
zidus76
Need help with a proof using factorials
Hey,

I need to prove that

(2n)! / n!(n+1)! is a natural number for every n∈N.

I tried already with induction, but got stuck.

Thanks.
• Oct 11th 2009, 05:14 PM
artvandalay11
Quote:

Originally Posted by zidus76
Hey,

I need to prove that

(2n)! / n!(n+1)! is a natural number for every n∈N.

I tried already with induction, but got stuck.

Thanks.

you need induction for this... where did you get stuck?
• Oct 11th 2009, 05:18 PM
zidus76
On the 3rd step, I plug in (n+1) and get

(2n+1)!/(n+1)!(n+2)!

Then I transform it to

(2n+1)2n!/n!(n+1)!(n+1)(n+2)

And since I assume that (2n)!/n!(n+1)! is a natural number I make it =k, then I got

k* (2n+1)/(n+1)(n+2)

If I could prove that (2n+1)/(n+1)(n+2) is also a natural number I would be done, but I can't do anything with (2n+1)/(n+1)(n+2).
• Oct 11th 2009, 05:20 PM
artvandalay11
yes sorry what you said cannot be proved via induction, you corrected your mistake

(i should read questions fully sorry)
• Oct 11th 2009, 05:28 PM
zidus76
Can't, since if you plug in 1 in the first step you get 1/2 and that's not a natural number. Anyway, I realized I made a computing mistake, it should be (2n+2)!, not (2n+1)!. Then I get (2n+1)(2n+2)/(n+1)(n+2), then I can cancel out (n+1) and then I get 2*(2n+1)/(n+2). But I still can't do anything with that, tried induction as well.

I also just read that these numbers are also known as the Catalan numbers, if it helps anyone. I couldn't find any proofs on the internet.