Fabonacci series

This is a simple numerical series which is first discovered by Leonardo Fibonacci in the 12 century.

This series begins from 0 and 1, and then next term one of this series equals the sum of the previous two numbers.

0,1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .

The Golden Ratio () is an irrational number with several curious properties. It can be defined as that number which is equal to its own reciprocal plus one: = 1/ + 1. Multiplying both sides of this same equation by the Golden Ratio we derive the interesting property that the square of the Golden Ratio is equal to the simple number itself plus one: 2 = + 1. Since that equation can be written as 2 - - 1 = 0, we can derive the value of the Golden Ratio from the quadratic equation, , with a = 1, b = -1, and c = -1: . The Golden Ratio is an irrational number, but not a transcendental one (like ), since it is the solution to a polynomial equation.

This gives us either 1.618 033 989 or -0.618 033 989. The first number is usually regarded as the Golden Ratio itself, the second as the negative of its reciprocal. The Golden Ratio can also be derived from trigonometic functions: = 2 sin 3/10 = 2 cos /5; and 1/ = 2 sin /10 = 2 cos 2/5. The angles in the trigonometric equations in degrees rather than radians are 54o, 36o, 18o, and 72o, respectively