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        riddles >> putnam exam (pure math) >> Combinatorial Sum
        
(Message started by: ThudanBlunder on Jan 20^th, 2009, 5:09am)

Title: Combinatorial Sum
Post by ThudanBlunder on Jan 20^th, 2009, 5:09am

49
Evaluate http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif(-1)^k http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lp.gif⁹⁹http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bfcc.gif_2khttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rp.gif = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lp.gif⁹⁹http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bfcc.gif₀http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rp.gif - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lp.gif⁹⁹http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bfcc.gif₂http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rp.gif + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lp.gif⁹⁹http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bfcc.gif₄http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rp.gif - ....... - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lp.gif⁹⁹http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bfcc.gif₉₈http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rp.gif
k=0

Title: Re: Combinatorial Sum
Post by towr on Jan 20^th, 2009, 7:50am

[hide]
1/2 sum_n=0..99 C(99, n) iⁿ 1^(99-n) + 1/2 sum_n=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
2^48.5 [exp(6*pi/8 i) + exp(10*pi/8 i)]
2^48.5 * -sqrt(2)
-2⁴⁹
[/hide]

Title: Re: Combinatorial Sum
Post by pex on Jan 20^th, 2009, 8:45am

So, more generally, sum_{k=0..floor(n/2)} (-1)^k ⁿC_2k =[hideb] 2^n/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
- 2^n/2 ........ if n = 4 mod 8.[/hideb]
Interesting!