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        riddles >> putnam exam (pure math) >> Factorial Sum
        
(Message started by: ThudnBlunder on Feb 15^th, 2011, 2:14pm)

Title: Factorial Sum
Post by ThudnBlunder on Feb 15^th, 2011, 2:14pm

49
Evaluate if you please http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif(-1)^k C(99,2k) = C(99,0) - C(99,2) + C(99,4) - ......... + C(99,96) - C(99,98)
k=0

Title: Re: Factorial Sum
Post by towr on Feb 16^th, 2011, 9:46am

[hide]
[ http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif⁹⁹ C(99,k) _k=0 i^k + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif⁹⁹_k=0 C(99,k) (-i)^k ]/2
[(1+i)⁹⁹ + (1-i)⁹⁹]/2
{[http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2 e^{pi/4 i}]⁹⁹ + [http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2 e^{-pi/4 i}]⁹⁹}/2
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2⁹⁷ [e^{99 pi/4 i} + e^{-99 pi/4 i}]
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2⁹⁷ [e^{3 pi/4 i} + e^{-3 pi/4 i}]
- http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2⁹⁸
- 2⁴⁹
[/hide]

Title: Re: Factorial Sum
Post by ThudnBlunder on Feb 16^th, 2011, 10:31am

Nicely done, towr. :)