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

Postby ana3a » Sat Mar 20, 2010 9:28 am

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

Postby jammidactyl » Sat Mar 20, 2010 12:22 pm

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

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

Re: Help with a finite sum

Postby lime » Sun Mar 21, 2010 7:17 am

\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}




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