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Math Help - Fibonacci sequence proof

  1. #1
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    Fibonacci sequence proof

    Hi,

    I'm having trouble solving the following proof on my study guide for an exam:

    Prove for the Fibonacci sequence F_k


    F_k*F_k - F_(k-1)*F_(k-1) = F_k*F_(k+1) - F_(k+1)*F_(k-1)

    for k>=1



    Could anyone show me what I need to be doing? Any help would be appreciated. Thanks!
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  2. #2
    MHF Contributor alexmahone's Avatar
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    F_k+F_{k-1}=F_{k+1}

    Multiplying both sides of the equation by F_k-F_{k-1},

    (F_k+F_{k-1})(F_k-F_{k-1})=F_{k+1}(F_k-F_{k-1})

    F_k^2-F_{k-1}^2=F_kF_{k+1}-F_{k+1}F_{k-1}

    F_kF_k-F_{k-1}F_{k-1}=F_kF_{k+1}-F_{k+1}F_{k-1}
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  3. #3
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    Quote Originally Posted by mattgavin View Post
    Hi,

    I'm having trouble solving the following proof on my study guide for an exam:

    Prove for the Fibonacci sequence F_k


    F_k*F_k - F_(k-1)*F_(k-1) = F_k*F_(k+1) - F_(k+1)*F_(k-1)

    for k>=1



    Could anyone show me what I need to be doing? Any help would be appreciated. Thanks!
    Alternatively, the Fibonacci recursion relation is

    F_{k+1}=F_k+F_{k-1}

    We are asked to prove F_kF_k-F_{k-1}F_{k-1}=F_kF_{k+1}-F_{k+1}F_{k-1}

    This contains F_{k-1} on the left, hence F_{k-1}=F_{k+1}-F_k

    Therefore, rewriting the left side

    F_kF_k-F_{k-1}F_{k-1}=F_kF_k-\left(F_{k+1}-F_k\right)\left(F_{k+1}-F_k\right)

    =F_kF_k-\left(F_{k+1}F_{k+1}-2F_{k+1}F_k+F_kF_k\right)

    =2F_kF_{k+1}-F_{k+1}F_{k+1}=F_kF_{k+1}+F_kF_{k+1}-F_{k+1}F_{k+1}

    F_{k+1} is a factor of the last 2 terms (it's a factor of all 3 of course), giving

    F_kF_{k+1}+F_{k+1}\left(F_k-F_{k+1}\right)=F_kF_{k+1}-F_{k+1}\left(F_{k+1}-F_k\right)

    In brackets is F_{k-1}
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  4. #4
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    Letting you know

    Just picked up a book on Fibonacci numbers by Alfred Posamentier (2007) which is interesting reading.
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