Comments on: Connecting Fibonacci and geometric sequences

By: Math At Play Blog Carnival #7 - Onomatopoeia

Math At Play Blog Carnival #7 - Onomatopoeia — Fri, 15 May 2009 10:02:23 +0000

[...] Cook presents Connecting Fibonacci and geometric sequences — The Endeavour posted at The [...]

By: Mark Reid

Mark Reid — Wed, 13 May 2009 00:21:49 +0000

@Wedge, that’s one way to prove it but the approach I had in mind doesn’t need to invoke an extra proposition about Fibonacci sequence (though that is yet another nice relationship between them and the golden ratio).

Notice that R_{n+1} := F_{n+1} / F_n = (F_n + F_{n-1}) / F_n = 1 + F_{n-1}/F_n = 1 + 1/R_n. The limit R of the sequence R_n is a fixed point of this recurrence. That is, R = 1 + 1/R and so R^2 = R + 1 which is exactly the equation defining φ.

By: Wedge

Wedge — Tue, 12 May 2009 20:04:46 +0000

@Mark the reason why the ratio of successive members of the Fibonacci series tends towards the golden ratio is because the sequence can be expressed as a linear combination of the above geometric sequences of phi and phi’. Specifically, there is an explicit formula for elements in the Fibonacci series, being: F(n) = phi^n/sqrt(5) – (-phi)^(-n)/sqrt(5) (this is also detailed in the “Fibonacci numbers at work” post), note that (-phi)^(-n) = (phi’)^n, so this translates to (phi^n – phi’^n)/sqrt(5)

Thus, the ratio of F(n+1) to F(n) is equal to (phi^(n+1) – phi’(^n+1))/(phi^n – phi’^n). Since abs(phi’) < 1, phi’^n will trend towards 0 as n gets larger, and this ratio will trend toward phi^(n+1)/phi^n, which is just phi.

By: Mark Reid

Mark Reid — Tue, 12 May 2009 03:26:19 +0000

Another, and perhaps more direct, relationship between the original Fibonacci sequence F_1, F_2, … and the golden ratio is that the limit of F_{n+1}/F_n as n goes to infinity is φ. This is quite easy to prove given the observation you have already made about the golden ratio.

Another beautiful property of the golden ratio that you can arrive at easily is that it has, in some sense, the simplest continued fraction expansion: [1; 1, 1, ...].