What are Fibonacci Numbers?
Fibonacci Numbers, named after Leonardo Fibonacci, is a sequence of numbers:
with a recurrence relation .
Relation to Golden Ratio
Fibonacci Numbers are linked to the mysterious Golden Ratio,
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:
Dividing throughout by , we get
(We will first assume exists for the time being, and prove it later)
Taking limits, we get .
Denoting as , we get:
Multiplying by , we get
This is a quadratic equation, solving using the quadratic equation, we get:
Since is clearly positive, we have which is the Golden Ratio!
For a complete proof, actually we will need to prove that 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.