Fibonacci series and golden ratio

[alexk307]

I was wondering if someone could explain the fibonacci series and golden ratio to me, I'm very curious, but I don't have that much experience in math as a high school trig student.

 

[Tac-Tics]

You might want to check out the wikipedia pages. These two have pretty good articles.

http://en.wikipedia.org/wiki/Golden_ratio
http://en.wikipedia.org/wiki/Fibonacci_number

The fibonacci sequence is pretty magical. It's definition is very simple

f(0) = 1
f(1) = 1
f(x) = f(x-1) + f(x-2), for all x >= 2

But is miraculously related to a ton of mathematical phenomena, especially in number theory.

 

[imranq]

Personally, I liked that $$\[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}} = \dfrac{1}{1 + \dfrac{1}{1+\frac{1}{1+...}}} = \phi\]$$ or the golden ratio. Also, note (since phi is defined by those recursive sequences) that $$\frac{1}{\phi} = \phi -1$$