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Summing the fiftieth power (Posted on 2012-11-29) Difficulty: 5 of 5
Find the sum of the fiftieth powers of all sides and diagonals of a regular 100-gon inscribed in a circle of radius R.

No Solution Yet Submitted by Danish Ahmed Khan    
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re: halfway through the general solution Comment 3 of 3 |
(In reply to halfway through the general solution by Michael Kornrade)

I repeat the two important formulas:
Sv = (2*R)^p * SUM( [sin( k * pi / N)]^p ) for k = 0...N-1
S = N * Sv / 2

For the moment, I cannot make the formula more compact, for p > 2...

Go to http://mathworld.wolfram.com/Sine.html to find compact formulas for p = 1 (eq.16 or eq.21) and p = 2 (eq.20).

So, in a Matlab computation for N = 100, p = 50 and R = 0.5 I get:
Sv =   11.2275172659217...
S  =   561.375863296086...

Edited on January 10, 2013, 7:09 pm

Edited on January 10, 2013, 7:12 pm
  Posted by Michael Kornrade on 2013-01-10 19:04:46

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