Details

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

while

.

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.