Generalized Fibonacci Sequence and the Golden Ratio
Generalized Fibonacci Sequence and the Golden Ratio
The sequence of Fibonacci numbers is given by 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …, in which each number is the sum of the two preceding numbers. As , the ratio =/ approaches , known as the golden ratio (or golden section or divine proportion), designated by .
n∞
ϕ
n
F
n
F
n-1
(1+
5
)2≈1.61803ϕ
A remarkable generalization of this result is that for an arbitrary pair of numbers and (not both zero), the generalized Fibonacci sequence :a,b,a+b,a+2b,2a+3b,3a+5b,5a+8b,… gives the same limiting ratio of successive members as , independent of the choices of and .
a
b
G
n
n∞
a
b