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Math Help - Fibonacci Proof by Induction

  1. #1
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    Fibonacci Proof by Induction

    Assuming F represents the Fibonacci sequence, use mathematical induction to prove that for all integers n >= 0, Fsub(n+2)Fsub(n) − (Fsub(n+1))^2 = (−1)^n.

    I have gotten to the inductive goal when plugging in k+1, but am lost from here.

    Thanks.
    Last edited by Walshy; November 14th 2012 at 01:13 PM.
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  2. #2
    Senior Member MacstersUndead's Avatar
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    Re: Fibonacci Proof by Induction

    Assuming, (after confirming base case)

    Fk+2Fk − (F(k+1)2 = (−1)k. (Assumption P)

    With the induction step, you want to prove

    Fk+3Fk+1 - (Fk+2)2 = (-1)k+1

    so,
    (-1)k+1 = (-1)k*(-1)^1 = -1 * ( Fk+2Fk − (F(k+1)2 ) (by Assumption P)

    Can you take it from here? I currently don't have pencil and paper handy to give you the full answer.

    You'll probably also have to use the fact that Fk+2 = Fk + Fk+1
    For simplicity, I would leave the bolded and then work in the other direction as well
    In other words, working with Fk+3Fk+1 - (Fk+2)2 so that it's equal to what is in the bold, since A = B = C implies A = C
    Thanks from Walshy
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    Re: Fibonacci Proof by Induction

    Yeah that makes sense, thanks. I'm in the process of trying to figure out the rest, will update.
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    Senior Member MacstersUndead's Avatar
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    Re: Fibonacci Proof by Induction

    I look forward to it. This problem did look very familiar from a number theory course I took.
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    Re: Fibonacci Proof by Induction

    Yeah i figured it out. Thanks for all the help.
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  6. #6
    Senior Member MacstersUndead's Avatar
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    Re: Fibonacci Proof by Induction

    No problem. By curiosity was there anything I missed noting that helped you solve the problem?
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  7. #7
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    Re: Fibonacci Proof by Induction

    No i don't think so. Your explanation was very thorough, and as u said, u didnt have a pencil and paper to work out the last part, but that wasn't too bad.
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