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September 25th, 2012, 10:47 AM  #1 
Newbie Joined: Sep 2012 Posts: 6 Thanks: 0  proving a property of the fibonacci sequence
Greetings everyone, I need to prove the following property: where means the term on the fibonacci's sequence. This is an exercise from a list where all the previous were solved using induction, so I expect the solution also goes through this way. 
September 25th, 2012, 11:28 AM  #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: proving a property of the fibonacci sequence
Do you know how to set up a proof by induction? Often the best way (at this level) is to start by filling in the skeleton of the proof, which makes it easier to find the rest. For example, suppose you wanted to prove that n + n = 2n by induction. The proof would look like this: For the base case, 1 + 1 = 2*1; the left side and right side are both 2, so this is true. Otherwise, suppose n > 1 and that (n1) + (n1) = 2(n1). Then [color=#00BFFF](inductive step here.)[/color] Thus n + n = 2n on this assumption. By induction, n + n = 2n for all n >= 1. Can you fill out the same for your problem? Once you do, make any simplifications you can from the inductive hypothesis. You might just solve it from there; if not, I'd be happy to help you fill in the gaps (unless someone beasts me to it). 
September 25th, 2012, 12:02 PM  #3 
Newbie Joined: Sep 2012 Posts: 6 Thanks: 0  Re: proving a property of the fibonacci sequence
yes, I do know how to use induction. We check the proposition for a first element then find a way to prove that assuming it works for n it shall work for n+1. The thing is, I am not being able to 'jump' from de n to the n+1 
September 25th, 2012, 12:07 PM  #4 
Newbie Joined: Sep 2012 Posts: 3 Thanks: 0  Re: proving a property of the fibonacci sequence
wow, I was trying to solve this exercise someday (using induction too). Assuming that it's true for k1, I couldn't reach that it would be true for k. I want this demonstration too, so I'll follow this topic. 10 points for whom answer this! 
September 25th, 2012, 12:08 PM  #5 
Newbie Joined: Sep 2012 Posts: 6 Thanks: 0  Re: proving a property of the fibonacci sequence
Normally when using iduction just some manipulation, even that some times a lot of manipulations, of one equation will get you from de nterm to the n+1term, bur here I don't see how to do thoese manipulations. Mainly, because on the left side we have while on the right . Those index are really kiling me. I didn't find a relation betwen then, no way to convert one into another.

September 25th, 2012, 06:36 PM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: proving a property of the fibonacci sequence
After having shown the base case to be true, state the hypothesis : (1) By the definition of the sequence, we have: Using the hypothesis, this is: (2) Adding (1) and (2), we have: We have derived from thereby completing the proof by induction. 
September 28th, 2012, 06:52 AM  #7  
Newbie Joined: Sep 2012 Posts: 6 Thanks: 0  Re: proving a property of the fibonacci sequence Quote:
anyway, thank you a lot Mark! This is just what I was looking for. As I expected, the solution is really simple(after you know her). Brilliant!  
September 28th, 2012, 07:30 AM  #8  
Newbie Joined: Sep 2012 Posts: 6 Thanks: 0  Re: proving a property of the fibonacci sequence Quote:
 
September 28th, 2012, 07:37 AM  #9 
Newbie Joined: Sep 2012 Posts: 6 Thanks: 0  Re: proving a property of the fibonacci sequence
I belive this invalidate the demonstration. by the way, is it possible to edit a post on the forum? 
September 28th, 2012, 08:56 AM  #10 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: proving a property of the fibonacci sequence
Since we have assumed the hypothesis to be true, we may use it in the inductive step. I would prefer not to of course, but I have seen this done before. Once your status changes from Newcomer, you will be able to edit posts. Perhaps we may alleviate the problem by observing: Continuing in this fashion, we will find: Now we may add this to the induction hypothesis. 

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