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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 jprime [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  
Math Team Joined: Apr 2012 Posts: 1,579 Thanks: 22  Re: Kinda trivial, but kinda neat! Quote:
 

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