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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.
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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…..
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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
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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.)
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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.