on 31.7.2005 1:27, Kirby Urner at firstname.lastname@example.org 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)