Introduction

Among numerical sequences, the Fibonacci numbers   have achieved a kind of celebrity status. These are referred to as one of the "two shining stars in the vast array of integer sequences”. The second of “shining stars" is the Lucas numbers, a close relative of the Fibonacci numbers. The Fibonacci Numbers are always being studied because of their complex nature.

The Fibonacci numbers are famous for possessing wonderful and amazing properties.  Some are well known. For example, the sums and differences of Fibonacci numbers are Fibonacci numbers, and the ratios of Fibonacci numbers converge to the golden mean.

Did you know that any four consecutive Fibonacci numbers can be combined to form a Pythagorean triple? Or how about this: The greatest common divisor of two Fibonacci numbers is another Fibonacci number.

With such fabulous properties, it is no wonder that the Fibonacci numbers stand out as a kind of super sequence. The purpose of this section is to demonstrate that many of the properties of the Fibonacci numbers can be stated and proved for a much more general class of sequences. We shall begin by reviewing a selection of the properties that made Fibonacci numbers famous. A number of the properties of the Fibonacci numbers will be seen to arise simply and naturally as the tools are presented.

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The Fibonacci Series

When a square is subtracted, another golden section is left behind.

No matter how many subdivisions , the ratio remains intact.

When the squares define the radii of circles, a familiar pattern emerges.