What is the connection between continued fractions, Fibonacci numbers, and the golden ratio?

Question

What is the connection between continued fractions, Fibonacci numbers, and the golden ratio?

Answer 1

A fibonacci sequence comes in the form F(n) = F(n-1) + F(n-2)

Now suppose you take the ratio of the nth term in a fibonacci sequence and the (n-1)th term in the same sequence. The limit of this ratio as n approaches infinity happens to be the golden ratio.



The really cool thing is this works for any two starting values, F(n-1) and F(n-2).



Taking the original fibonacci sequence as an example, the visual representation of this is as follows:



Fibonacci Sequence:



1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...



The first ratio is 1/1

Second ratio 2/1

Third is 3/2

Fourth is 5/3

And so on. This ratio alternates above and below the golden ratio, but can be shown to converge to it. Mathematically:



limit(n-->inf) of F(n+1)/F(n) = golden ratio

Answer 2

The continued fraction of the golden ratio has convergents which are ratios of consecutive Fibonacci numbers.

Answer 3

Fibonacci numbers and phi or Phi

If you look at the ratio of each number to the next number, it gets closer and closer to phi:

1, 1 ==> 1

1, 2 ==> .5

2, 3 ==> .6666

3, 5 ==> .6

5, 8 ==> .625

8, 13 ==> .61

and so on.



as for continued fractions, Phi is easy to get