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Math Help - Fibonacci sequence identity

  1. #1
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    Fibonacci sequence identity

    Hi, we've been trying to prove this identity and can't quite see how its done. Any insights would be greatly appreciated. . .

     \text{where } \newline F_1 = 1 \newline F_2 = 1, \newline F_n = F_{n-1} + F_{n-2} \text{ for } n \geq 3 \newline \newline L_1 = 1 \newline L_n = F_{n+1} + F_{n-1}   \text{ for } n \geq 2

    show that

    F_{2n} = F_n L_n

    Thanks, MD
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  2. #2
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    Re: Fibonacci sequence identity

    Quote Originally Posted by Mathsdog View Post
    Hi, we've been trying to prove this identity and can't quite see how its done. Any insights would be greatly appreciated. . .

     \text{where } \newline F_1 = 1 \newline F_2 = 1, \newline F_n = F_{n-1} + F_{n-2} \text{ for } n \geq 3 \newline \newline L_1 = 1 \newline L_n = F_{n+1} + F_{n-1}   \text{ for } n \geq 2

    show that

    F_{2n} = F_n L_n

    Thanks, MD
    The slick way to do this is as follows. Define a 2x2 matrix A by A = \begin{bmatrix}1&1\\1&0 \end{bmatrix}. Prove by induction that A^n = \begin{bmatrix}F_{n+1}&F_n\\F_n&F_{n-1} \end{bmatrix}. Then look at the off-diagonal elements of the matrices A^n\times A^n and A^{2n} (which are equal, of course).
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