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Math Help - Fibonacci Proof

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    Fibonacci Proof

    Prove the following identity:
    (F(n+1))^2 - (F(n))^2 = F(n-1)F(n+2)
    I am trying to do this with induction, and I am stuck!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by veronicak5678 View Post
    Prove the following identity:
    (F(n+1))^2 - (F(n))^2 = F(n-1)F(n+2)
    I am trying to do this with induction, and I am stuck!
    Merely note that F_{n+1}^2-F_n^2=\left(F_{n+1}-F_n\right)\left(F_{n+1}+F_n\right). Noting then that F_{n+2}=F_{n+1}+F_{n} and F_{n+1}=F_n+F_{n-1} gives the desired result.
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    Quote Originally Posted by veronicak5678 View Post
    Prove the following identity:
    (F(n+1))^2 - (F(n))^2 = F(n-1)F(n+2)
    I am trying to do this with induction, and I am stuck!
    Alternatively (using induction to practice! given the clarity of Drexel's response) if

    \left(F_{n+1}\right)^2-\left(F_n\right)^2=F_{n-1}F_{n+2}

    then we require that

    \left(F_{n+2}\right)^2-\left(F_{n+1}\right)^2=F_nF_{n+3}

    Proof

    F_nF_{n+3}=F_n\left(F_{n+1}+F_{n+2}\right)=F_nF_{n  +1}+F_nF_{n+2}

    Therefore, is

    \left(F_{n+2}\right)^2-\left(F_{n+1}\right)^2=F_nF_{n+1}+F_nF_{n+2}\;\;?

    \left(F_{n+2}\right)^2-F_nF_{n+2}=\left(F_{n+1}\right)^2+F_nF_{n+1}\;\;?

    F_{n+2}\left[F_{n+2}-F_n\right]=F_{n+1}\left[F_{n+1}+F_n\right]\;\;?

    F_{n+2}=F_{n+1}+F_n\Rightarrow\ F_{n+2}-F_n=F_{n+1}

    gives

    F_{n+2}F_{n+1}=F_{n+1}F_{n+2}\;\;?

    hence the hypothesis is true
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