Question

How does the golden ratio relate to the Fibonacci sequence?

Answer 1

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)`