# How does the golden ratio relate to the Fibonacci sequence?

Nov 7, 2015

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 $1.618033988 \ldots$

An example is the division $34 \div i \mathrm{de} 21$, which gives the answer $1.61904762$. Going even further to $17711 \div i \mathrm{de} 10946$, which gives $1.61803399$.

In addition, there is a direct formula for ${F}_{n}$ in terms of $\phi$:

${F}_{n} = \frac{{\phi}^{n} - {\left(- \phi\right)}^{-} n}{\sqrt{5}}$