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Thread: Induction and Fibonacci Numbers

  1. #1
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    Smile Induction and Fibonacci Numbers

    I am trying to figure out how to show that $\displaystyle f_{n+1}f_{n-1}-f_{n}^{2}=(-1)^{n}$ when n is a positive integer.

    My Work
    P(n) is $\displaystyle f_{n+1}f_{n-1}-f_{n}^{2}=(-1)^n$ for $\displaystyle n>=1$

    Basis step - P(1) is true because $\displaystyle f_{2}f_{0}-f_{1}^{2}=1*0-1=-1=(-1)^1$
    Inductive step - Assume $\displaystyle P(k)=f_{k+1}f_{k-1}-f_{k}^{2}=(-1)^k$ is true,
    then $\displaystyle P(k+1)=f_{k+2}f_{k}-(f_{k+1}^2=(-1)^{k+1} $is true

    Then prove $\displaystyle P(k+1)=[f_{k+1}f_{k-1}-f_{k}^{2}]+[f_{k+2}f_{k}-f_{k+1}^{2}]=(-1)^k+[f_{k+2}f_{k}-f_{k+1}^2]$, but I can't figure out how to get this P(k+1) equation to equal $\displaystyle (-1)^{k+1} $

    I received the reply below, which is probably a wonderful answer, but I am unclear as to where in the 3rd line came from. I though you had to add the initial part of the P(k+1) equation, which is $\displaystyle f_{k+1}f_{k}-f_{k+1}^2$, to both sides of the P(k) equation as I did above.

    First Reply
    Assume for n.

    We want to show that





    ,<--- negative of the induction hypothesis

    by the induction hypothesis.



    So, is true.
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  2. #2
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    $\displaystyle f_n(f_{n+1}+f_n) - f_{n+1}(f_{n}+f_{n-1})$ comes directly from the definition of the Fibonacci numbers. Remember that $\displaystyle f_n = f_{n-1} + f_{n-2}$?
    So in your case in $\displaystyle f_{n}f_{n+2} - f^2_{n+1}$
    $\displaystyle f_{n+2} = f_{n+1}+f_{n}$
    and
    $\displaystyle f^2_{n+1} = f_{n+1}f_{n+1} = f_{n+1}(f_{n}+f_{n-1}) $,
    since $\displaystyle f_{n+1} = f_{n} + f_{n-1}$
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  3. #3
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    Question

    The math makes sense but I still do not understand why you are allowed to do the "negative of the inductive hypothesis" to get the extra -1 to make the answer $\displaystyle {-1}^{K}$. Doesn't the answer also still need to be $\displaystyle {-1}^{K+1}$?

    Also, I thought it wasn't supposed to prove the exact P(k+1) equation in the inductive step but rather the P(k) equation with the first part of P(k+1) added to both sides as follows:



    Then, aren't I supposed to work on the second half of this equation to make it equal $\displaystyle {-1}^{K+1}$?


    EDIT:

    The math makes sense but I still do not understand why you are allowed to do the "negative of the inductive hypothesis" to get the extra -1 to make the answer be ? I already understand that -$\displaystyle {-1}^{k}={-1}^{k+1}$ I just don't know where the exra -1 came from.
    Last edited by mr fantastic; Jan 5th 2009 at 06:07 AM. Reason: Merge
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  4. #4
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    First of all
    $\displaystyle (-1)^k \not= (-1)^{k+1}$
    because they have opposite signs.

    Second of all, extra $\displaystyle -1$ comes from swapping terms in the equation of the inductive assumption, as in $\displaystyle 5-2=3, but 2-5=-(3)$. That's it.
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