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Math Help - Fibonacci and golden section proof

  1. #1
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    Fibonacci and golden section proof

    hello,

    based on the formular for Fibonacci numbers I am supposed to calculate the value for the golden section phi.
    (1) rn = Fn+1 / Fn converges against phi, and I have to prove that
    (2) rn+1 = 1 + 1 / rn and thus phi = 1 + 1 / phi

    But how do I get from (1) to (2) ?

    any help greatly appreciated!

    oz.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ozfingwe View Post
    hello,

    based on the formular for Fibonacci numbers I am supposed to calculate the value for the golden section phi.
    (1) rn = Fn+1 / Fn converges against phi, and I have to prove that
    (2) rn+1 = 1 + 1 / rn and thus phi = 1 + 1 / phi

    But how do I get from (1) to (2) ?

    any help greatly appreciated!

    oz.
    There is a "better" way to do this, if you are curious. But just note that r_{n+1}=\frac{F_{n+2}}{F_{n+1}}=\frac{F_{n+1}+F_{n  }}{F_{n+1}}=1+\frac{F_n}{F_{n+1}}=1+\frac{1}{r_n}
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  3. #3
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    just what I had in mind and couldn't put onto paper, thanks a lot mate!
    oz.
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