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Accuracy, Fibonacci + Golden Ratio

  1. Aug 10, 2008 #1
    I have been curious about this for a while...

    I'm interested to know if there is any easy way to tell the accuracy of the (n+1)th on the nth term of the Fibonacci series in relation to the golden ratio.

    I know that as n tends to infinity the ratio tends to the Golden Ratio "Phi" - but is there a way to tell, say, to how many decimal places the 32nd on the 31st term is close to Phi?
  2. jcsd
  3. Aug 10, 2008 #2
    We can do even better, and give Binet's closed-form expression for the [itex]n^{th}[/itex] Fibonacci number in terms of the golden ratio [itex]\phi[/itex]:

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

    Sorry to give away so much, but mathematics is large enough :)
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