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Math Help - fibonacci and the golden ratio proof

  1. #1
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    fibonacci and the golden ratio proof

    any help on getting started would be great. i really don't know where to go with this as i'm useless with limits.

    alpha is the golden ration ((1+sqrt5)/2)

    f2k etc are part of the fibonacci sequence

    prove lim f2k/f2k+1 = 1/alpha
    k->infinity
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  2. #2
    A Plied Mathematician
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    The Fibonacci sequence can be written as a recurrence relation. Solve that relation for the nth term, and then form the desired ratio and take the limit. Does that give you some ideas?
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  3. #3
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    Here's a hint: The golden ratio is the solution to the equation x^2-x-1=0

    Now look at the recurrence relationship of Fibonacci numbers, can you find this equation?
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  4. #4
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by gpenguin View Post
    any help on getting started would be great. i really don't know where to go with this as i'm useless with limits.

    alpha is the golden ration ((1+sqrt5)/2)

    f2k etc are part of the fibonacci sequence

    prove lim f2k/f2k+1 = 1/alpha
    k->infinity
    The Fibonacci's sequence is the solution of the 'recursive relation'...

    \displaystyle y_{n} = y_{n-1} + y_{n-2}\\, y_{0}=0\\, \\y_{1}=1 (1)

    The (1) is linear and homogenous and its general solution is...

    \displaystyle y_{n} = c_{1}\ \varphi^{n} + c_{2}\ (-\varphi)^{-n} (2)

    ... where \varphi = \frac{1+\sqrt{5}}{2} is the 'golden ratio'. Now in (2) is...

    \displaystyle \lim_{n \rightarrow \infty} (-\varphi)^{-n}=0 (3)

    ... so that is...

    \displaystyle \lim_{n \rightarrow \infty} \frac{y_{n}}{y_{n-1}} = \varphi (4)

    Kind regards

    \chi \sigma
    Last edited by chisigma; February 9th 2011 at 01:09 PM.
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