# Fibonacci Numbers and the Mysterious Golden Ratio

## What are Fibonacci Numbers?

Fibonacci Numbers, named after Leonardo Fibonacci, is a sequence of numbers:

$F_0=0, F_1=1, F_2=1, F_3=2, F_4=3, F_5=5$,

with a recurrence relation $F_n=F_{n-1}+F_{n-2}$.

## Relation to Golden Ratio

Fibonacci Numbers are linked to the mysterious Golden Ratio, $\displaystyle \phi=\frac{1+\sqrt{5}}{2}\approx 1.61803$

In fact, the ratio of successive Fibonacci numbers converges to the Golden Ratio! The first person to observe this is Johannes Kepler.

How do we prove it?

Recall the recurrence relation: $F_n=F_{n-1}+F_{n-2}$

Dividing throughout by $F_{n-1}$, we get $\displaystyle \frac{F_n}{F_{n-1}}=1+\frac{F_{n-2}}{F_{n-1}}$

(We will first assume $\displaystyle\lim_{n\to\infty}\frac{F_n}{F_{n-1}}$ exists for the time being, and prove it later)

Taking limits, we get $\displaystyle\lim_{n\to\infty}\frac{F_n}{F_{n-1}}=1+\lim_{n\to\infty}\frac{F_{n-2}}{F_{n-1}}$.

Denoting $\displaystyle\lim_{n\to\infty}\frac{F_n}{F_{n-1}}$ as $\phi$, we get:

$\displaystyle \phi=1+\frac{1}{\phi}$

Multiplying by $\phi$, we get $\phi^2=\phi +1$

$\phi^2-\phi-1=0$

This is a quadratic equation, solving using the quadratic equation, we get:

$\displaystyle \phi=\frac{1\pm\sqrt{1^2-4(1)(-1)}}{2}=\frac{1\pm\sqrt{5}}{2}$

Since $\phi$ is clearly positive, we have $\displaystyle \phi=\frac{1+\sqrt{5}}{2}$ which is the Golden Ratio!

For a complete proof, actually we will need to prove that $\displaystyle\frac{F_n}{F_{n-1}}$ converges. This is a bit tricky and requires some algebra.

Interested readers can refer to the excellent website at: http://pages.pacificcoast.net/~cazelais/222/fib-limit.pdf

for more details.

Interesting video on Fibonacci numbers!

Fibonacci numbers and the Golden Ratio can also be used for trading stocks.

## Author: mathtuition88

https://mathtuition88.com/

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