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 .[more]
A remarkable generalization of this result is that for an arbitrary pair of numbers and (not both zero), the generalized Fibonacci sequence gives the same limiting ratio of successive members as , independent of the choices of and .[less]
Consider the original Fibonacci sequence: divide the recursion relation by to give
With some work, you can prove that the ratios form a Cauchy sequence, so the sequence has a limit; call it . Thus
This gives or , a quadratic equation with roots
The positive root
gives the golden ratio.
For the generalized Fibonacci sequence , the general term can be written using Fibonacci numbers as . Therefore,
In the limit , this approaches
independent of or , as long as not both are zero.