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#238 maths! http://www.timefan.com/viewtopic.php?f=15&t=3606 
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Author:  Spider [ December 24th, 2010 09:23:54 am ] 
Post subject:  #238 maths! 
The sum of the first eight Fibonacci numbers is equal to the tenth Fibonacci number minus one. Exercise: Prove the generalized form of this claim. Generalised (because I'm British) claim: The sum of any 8 consecutive Fibonacci numbers n1  n8 is equal to the Fibonacci number n10  n2. I could notate this more clearly if only I could be bothered to render it in LaTeX :P Proof: Lets call the first in our set of consecutive numbers a, and the second b. Then we have the sequence: Code: n1 n2 n3 n4 n5 n6 n7 n8 n9 n10 a b a+b a+2b 2a+3b 3a+5b 5a+8b 8a+13b 13a+21b 21a+34b The sum of the first 8 of these is 21a+33b. The difference then between sum(n1:n8) and n10 is b, which is exactly n2, and thus sum(n1:n8) is n10  n2 
Author:  william [ December 26th, 2010 07:55:18 am ] 
Post subject:  Re: #238 maths! 
The other generalization:http://dl.dropbox.com/u/4386762/fibonacci.pdf 
Author:  Spider [ January 4th, 2011 14:53:52 pm ] 
Post subject:  Re: #238 maths! 
Lovely stuff, I wish I had bothered to lay mine out like that... :) I wonder which of the two generalisations Ben had in mind? 
Author:  Diane Heaton [ January 4th, 2011 22:33:43 pm ] 
Post subject:  Re: #238 maths! 
William's is the one I had in mind, but both answers are acceptable. 
Author:  Token [ January 20th, 2011 15:50:47 pm ] 
Post subject:  Re: #238 maths! 
Of course, they both generalize to the claim that the sum of the mth to nth Fibonnaci numbers is the (n+2)th Fibonacci number minus the (m+1)th. 
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