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Topic: Which Polynomial?
Replies: 28   Last Post: Jul 27, 2006 3:13 PM

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Posts: 59
Registered: 1/30/05
Re: Which Polynomial?
Posted: Jul 31, 2005 9:07 PM
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on 31.7.2005 1:27, Kirby Urner at wrote:

> What converges to phi, as the Fibonacci's diverge, is the
> ratio between f[n+1] and f[n]. Whether we want to jump
> from here to your generative polynomial piece would be
> at student/teacher disgression. Definitely it's a
> connected topic. There's also going from continued
> fractions to Diophantine equations, as discussed
> previously (a few weeks ago), and to interesting
> representations of roots. It's a vast network, and the
> trick is to pick an intelligible path that (a) hangs
> together (b) tells a good story (isn't too boring) (c)
> contains life-relevant skill building.

The convergence is quite easily derived. With Phi = (1+SQRT5)/2, as n goes
to infinity, |(1-SQRT5)/2|^n (<1) goes to zero so lim[f(n+1)/f(n)] for n ->
inf is Phi.

It's indeed a very rich topic (multiple points of entry -> connections)


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