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 May 22nd, 2013, 12:22 PM #1 Math Team   Joined: Apr 2012 Posts: 1,579 Thanks: 22 Kinda trivial, but kinda neat! While gazing at and admiring the j-prime [1,2,2,3,5], I noticed that it was a very near miss to the fibonacci sequence 1,1,2,3,5 and that the sum of the first 4 members are the next fibonacci number 8 and the sum of all five is 13, the next one yet. This led me to conjecture that the sum of the first n fibonacci numbers is 1 less than the n+2nd fibonacci number. So, the "sum" of the first fibonacci number is 1, which is indeed 1 less than the 3rd fibonacci number, ie 2. So assume that the sum of the first n fibonacci numbers is if the n+2 fibonacci number minus 1. That means that the sum of the first n+1 fibonacci numbers equals the the sum of the first n + the n+1st fibonacci number. Ok, so the n+3rd fibonacci number is the sum of the n+1st + n+2nd fibonacci numbers. By assumption the n+2nd fibonacci number minus 1. So the sum of the the first n+1 fibonaccie numbers equals the n+1st fibonnaci number plus the n+2nd minus 1. Meanwhile, the n+3rd fibonacci number equals the n+1st plus n+2nd fibonacci number. So the sum of the first n+1 fibonacci numbers is one less than then n+3rd, ie the (n+1)+2nd fibonacci number, completing the induction. I'm sure this is a long since known result, but it was fun running into it unexpectedly and working it out! May 22nd, 2013, 12:29 PM #2 Global Moderator   Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Kinda trivial, but kinda neat! Indeed. Usually now I just look up these sorts of sequences in the OEIS; it's faster than doing the quick induction.  May 22nd, 2013, 12:33 PM   #3
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Re: Kinda trivial, but kinda neat!

Quote:
 Originally Posted by CRGreathouse Indeed. Usually now I just look up these sorts of sequences in the OEIS; it's faster than doing the quick induction. Ah yes, but some of us need a little practice!  Tags kinda, neat, trivial Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post beesee Applied Math 4 March 3rd, 2013 10:28 AM Bucephalus Linear Algebra 10 January 28th, 2012 09:08 PM pyth0n Calculus 3 September 28th, 2011 08:54 PM Agno Number Theory 4 July 19th, 2011 11:05 AM Pmaleymia New Users 7 September 25th, 2009 06:15 AM

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