How is the Golden Ratio and the Fibonacci sequence related?

Question

How is the Golden Ratio and the Fibonacci sequence related?

Answer 1

Fibonacci sequence is as followed

0, 1, 1, 2, 3, 5, 8, 13, 21.....and so on.

Golden ratio is (1+sqrt(5))/2

approx 1.61803

Starting with the 5th term of the Fibonacci and divide it by its predecessor,  the 4th term and so on to n/n-1( where n is a fibonacci number and n-1 is the previous fibonacci number) you will get approximatly the golden ratio of 1.6. The numbers will be higher and lower than the actual ratio but will converge the more numbers you put in the sequence.

Answer 2

Fibonacci sequence is defined by recurrence equation

f(n+2) = f(n+1) + f(n)

Lets find expression for f(k) in closed form as a function of k in form

f(k) = a^k

Substitution yields:

a^(k+2) = a^(k+1) + a^k

a^k (a^2 - a  - 1) = 0

a^2 - a  - 1 = 0

The solution of the above written equation

a = (1+√5)/2

is called golden ration, and consequently

limit as k tends to +oo of a(k+1)/a(k) = golden ratio.

Answer 3

The ratio of a number in the Fibonacci series OVER the previous becomes increasingly closer to the golden ratio.