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Need help w/ Fibonnaci and Golden Ratio proof

  1. Oct 17, 2011 #1
    Let F_n and F_n+1 be successive Fibonnaci numbers. Then |(F_n+1)/(F_n) - Phi | < 1/(2(F_n)^2)

    Where Phi is the Golden ratio.
  2. jcsd
  3. Oct 18, 2011 #2
    Just use the formula for nth fibonacci number.
  4. Oct 18, 2011 #3
    It's not that simple. I tried fracturing it many times but every time it just seem to leave to the same result.
  5. Oct 18, 2011 #4
    Remember that the nth Fibonacci number is:

    [tex]F_n = \frac {1} {\sqrt{5}} \left [ \phi^n - \left ( \frac {-1}{\phi}\right)^n\right][/tex]

    and you can multiply Fn2 through your equation:

    [tex] \left | F_{n+1} \times F_{n} - \phi \times F^2_n \right | < \frac {1}{2} [/tex]

    What happens when you set the whole equation to terms of [itex]\phi[/itex]?

    Note that for the first 3 Fibonacci numbers: 0,1,1 the formula doesn't work. Additionally, as we get deeper into the Fibonacci sequence, [itex]F_n[/itex] as [itex]n \to \infty[/itex], what does [itex] \left | F_{n+1} \times F_{n} - \phi \times F^2_n \right |[/itex] approach? hint: you might see the value first for a specific very low n
    Last edited: Oct 18, 2011
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