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I Fibonacci series

  1. Sep 23, 2016 #1
    I'm not sure this is the right forum, so if not, please move to the appropriate forum. My question is why does the ratio of two consecutive fibonacci numbers converge to the golden ratio? I see no mathematical connection between the series ratios and ratios of a unit line segment divided into two unequal segments. .
     
  2. jcsd
  3. Sep 23, 2016 #2

    mfb

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    2016 Award

    Staff: Mentor

    You can show that
    (a) if two positive real numbers have this ratio, then their sum and the larger number have this ratio again
    (b) if the ratio is larger, then the ratio of the sum and the larger number is smaller, and vice versa
    (c) that the difference to the golden ratio always decreases in the cases of (b)
     
  4. Sep 23, 2016 #3

    fresh_42

    Staff: Mentor

    The Fibonacci sequence is defined by ##F(n+1)=F(n)+F(n-1)## and therefore
    $$ \frac{F(n+1)}{F(n)}= \frac{F(n)+F(n-1)}{F(n)} = 1 + \frac{F(n-1)}{F(n)} $$
    and the golden ratio is defined by
    $$ \frac{a+b}{a}=\frac {a}{b} =: \varphi $$
    If the Fibonacci sequence converges to ##\Phi##, then ##\Phi = 1 + \frac{1}{\Phi}##.

    You may try and find out why ##\Phi = \varphi## or read the following passage:
    https://en.wikipedia.org/wiki/Golden_ratio#Relationship_to_Fibonacci_sequence
     
  5. Sep 23, 2016 #4
    Thank you gentlemen. I guess I should have searched the internet first,
     
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