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
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.
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
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
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When a square
is subtracted, another golden section is left behind.
No matter how
many subdivisions , the ratio remains intact.
squares define the radii of circles, a familiar pattern emerges.