"Fibonacci" numbers are defined recursively:

F(0) = 0, F(1) = 1
F(n+1) = F(n) + F(n-1) for n >=1

Claim:

As n approaches infinity, F(n+1) / F(n) approaches the "golden ratio."
Draw a point on a "unit line" such that the ratio of the longer segment
to the shorter segment equals the reciprocal of the longer segment. This
ratio is called the "golden ratio." It is architecturally pleasing
(Parthenon) and naturally occurring (sunflowers).



Sunflower head displaying spirals of 21 (clockwise) and 34 (counterclockwise)


-------------------.----------     this is a "unit line" since d + (1 - d) = 1
        d             1 - d

d / (1 - d) =  1 / d   see above discussion of point which defines golden ratio 
(1 - d ) / d = d       take reciprocals

Calling 1 / d = g, we have

g - 1 = g^(-1)         substitution and algebra
g^2 - g - 1 = 0        this is a "quadratic equation" w/ a=1, b=-1 and c=-1

As n -> infinity

F(n+1) / F(n) -> r              see above claim
(F(n) + F(n-1)) / F(n) -> r     see above definition of Fibonacci numbers
1 + F(n-1) / F(n) -> r          algebra
1 + r^(-1) = r                  since F(n-1) / F(n) -> r^(-1) 
r + 1 = r^2                     algebra

r = g                  the quadratic equations in g and r are identical 

g = (-(-1) + ((-1)^2 -4(1)(-1))^.5) / 2(1)  use the "quadratic formula"
g = (1 + 5^.5) / 2
g = approx. 1.6180339887498948

It turns out (where phi = our g)





There is a connection between the Mandelbrot set and Fibonacci numbers.