TimeFanhttp://www.timefan.com/ #238 maths!http://www.timefan.com/viewtopic.php?f=15&t=3606 Page 1 of 1

 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 :PProof: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        n10a   b   a+b   a+2b   2a+3b   3a+5b   5a+8b   8a+13b   13a+21b   21a+34bThe 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.