perplexus.info :: Just Math : The golden ratio

The golden ratio (Posted on 2015-09-16)

In a Fibonacci sequence 1, 1, 2, 3, 5, …, F_n, F_n+1
define R_n = F_n/ F_n-1

Prove that lim (R_n) 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)

Comments: ( Back to comment list | You must be logged in to post comments.)

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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