How does the golden ratio relate to the Fibonacci sequence?

1 Answer

The ratio between successive terms of the Fibonacci sequence tends towards #phi#, or the golden ratio.

Explanation:

If you take two consecutive terms in the Fibonacci Sequence, or indeed any given sequence, and divide one of the terms by the one that comes before it, then you get a number that will get closer and closer to #phi#, or #1.618033988...#

An example is the division #34 divide 21#, which gives the answer #1.61904762#. Going even further to #17711 divide 10946#, which gives #1.61803399#.

In addition, there is a direct formula for #F_n# in terms of #phi#:

#F_n = (phi^n-(-phi)^-n)/sqrt(5)#