Thursday, 8 March 2012

Geometric Fibonacci is Golden

A Fibonacci sequence is a sequence of numbers {Fn} satisfying the recurrence relation, for n > 2,

F_{n} = F_{n-1}+F_{n-2} \,

with the first two terms F1 and F2 both being 1.

A geometric sequence is a sequence of numbers {an} satisfying the recurrence relation, for n > 1,

a_n = r\,a_{n-1}

with the first term a1 and r being fixed numbers.

If we set the first two terms of the Fibonacci sequence F1 = 1 and F2φ, and make it a geometric sequence with rφ, that is, for n > 1,

\varphi^n  = \varphi^{n-1} + \varphi^{n-2}\,

then it can be easily proved that, for φ > 0,

\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\dots\,

which is the golden ratio.