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}.

Fibonacci.jpg
Fibonacci

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.