Fibonacci numbers are named after Leonardo Fibonacci. In 1200, he wrote a book denoting and explaining these numbers. However, Indian mathematician Gopala and Hemachandra were the first to use them 50 years earlier.
Concept of the Fibonacci Numbers
The first two numbers of the Fibonacci number series are 0 and 1. Afterwards, the numbers are obtained by adding the previous two numbers like below:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34…..
Formula for Calculating (n+1)th Fibonacci Term Directly
We have a straight forward relation for calculating the (n+1)th term of the Fibonacci series as below:
F(n+1)= [(phi)^n – (-phi)^n] / SQRT(5)…………………..1.1
F(n+1)=(n+1)th Fibonacci series number
Phi = [1+SQRT(5)]/2
If for the Fibonacci number series you write the ratios of the (n+1)th term to nth term, where n starts from 2, you will get:
So, after first few numbers of the Fibonacci number series, this ratio tends to become constant and called golden ratio or golden section or phi and its value is:
Golden ratio (phi) = [1+SQRT(5)]/2 =1.618 (approx.)
The golden ratio (phi) can be seen in nature. If you observe the middle portion of a sunflower, the numbers of seed in one circle to the next circle follow the golden ratio. It has been said that for a perfect human body, the distance from the naval to foot and from the naval to head follow the golden ratio.