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The golden ratio (Posted on 2015-09-16) Difficulty: 3 of 5
In a Fibonacci sequence 1, 1, 2, 3, 5, …, Fn, Fn+1
define Rn = Fn/ Fn-1

Prove that lim (Rn) as n approaches infinity
is .5*(1+sqrt(5))=1.618...
a.k.a. the golden ratio, φ (phi).

See The Solution Submitted by Ady TZIDON    
Rating: 4.0000 (1 votes)

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Discretion helps | Comment 4 of 6 |
I had trouble proving this using normal analysis, but it follows simply if using the closed discrete formula for F(n)

F(n) = (Phi^n - (-1/Phi)^n)/sqrt(5)

Then R(n) = (Phi^n - (-1/Phi)^n)/(Phi^(n-1) - (-1/Phi)^(n-1))

As n goes to infinity, the (-1/Phi)^n terms go to zero, so
the limit of R(n) as n goes to infinity exists and equals the limit as n goes to infinity of Phi^n/(Phi^(n-1), which is Phi

  Posted by Steve Herman on 2015-09-18 12:56:14
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