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Topic: Combinatorial Sum (Read 1592 times) |
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ThudnBlunder
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Combinatorial Sum
« on: Jan 20th, 2009, 5:09am » |
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49
Evaluate  (-1)k  99 2k = 990 - 992 + 994 - ....... - 9998
k=0
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towr
wu::riddles Moderator
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Re: Combinatorial Sum
« Reply #1 on: Jan 20th, 2009, 7:50am » |
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1/2 sumn=0..99 C(99, n) in 1(99-n) + 1/2 sumn=0..99 C(99, n) (-i)n 1(99-n)
((1+i)99+(1-i)99)/2
[sqrt(2)99 exp(99 * 2pi * 1/8 i) + sqrt(2)99 exp(99 * 2pi * 7/8 i)]/2
248.5 [exp(6*pi/8 i) + exp(10*pi/8 i)]
248.5 * -sqrt(2)
-249
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| « Last Edit: Jan 20th, 2009, 7:52am by towr » |
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pex
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Re: Combinatorial Sum
« Reply #2 on: Jan 20th, 2009, 8:45am » |
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So, more generally, sumk=0..floor(n/2) (-1)k nC2k =| hidden: | 2n/2 ......... if n = 0 mod 8
2(n-1)/2 ... if n = +- 1 mod 8
0 ............. if n = +- 2 mod 8
- 2(n-1)/2 ... if n = +- 3 mod 8
- 2n/2 ........ if n = 4 mod 8. |
Interesting!
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