## Help with a finite sum

Forum for the GRE subject test in mathematics.
ana3a
Posts: 26
Joined: Fri Mar 13, 2009 6:12 am

### Help with a finite sum

Is there a simple formula for this sum?

$\sum_{k=0}^nk(n-k)=0\cdot n+1\cdot (n-1)+2\cdot(n-2)+\cdots+(n-1)\cdot 1+n\cdot 0$

It looks kind of familiar .. something to prove with induction in a proof writing class.
Thanks

jammidactyl
Posts: 6
Joined: Sat Oct 10, 2009 7:41 pm

### Re: Help with a finite sum

Distribute the k, break into difference of two sums... see what you can do.

lime
Posts: 129
Joined: Tue Dec 04, 2007 2:11 am

### Re: Help with a finite sum

$\sum_{k=0}^{n}k(n-k)=n\sum_{k=0}^{n}k-\sum_{k=0}^{n}k^2=\frac{n^2(n+1)}{2}-\frac{n(n+1)(2n+1)}{6}=\frac{(n-1)n(n+1)}{6}$