Interesting Properties of Fibonacci sequence

The Fibonacci series or sequence is a set of numbers that starts with a one or a zero, followed by a one, and proceeds based on the rule that each number (called a Fibonacci number) is equal to the sum of the preceding two numbers. If the Fibonacci sequence is denoted F (n), where n is the first term in the sequence, the following equation obtains for n = 0, where the first two terms are defined as 0 and 1 by convention:

F (0) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ...

In some texts, it is customary to use n = 1. In that case the first two terms are defined as 1 and 1 by default, and therefore:

F (1) = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ...

The Fibonacci sequence is named for Leonardo Pisano (also known as Fibonacci), an Italian mathematician who lived from 1170 – 1250. He used the arithmetic series in 1202 to illustrate a problem based on a pair of breeding rabbits, although the sequence had been described earlier in Indian mathematics.

It is a deceptively simple series, but its ramifications and applications are nearly limitless. It has fascinated and perplexed mathematicians for over 700 years, and nearly everyone who has worked with it has added a new piece to the Fibonacci puzzle, a new tidbit of information about the series and how it works. Fibonacci mathematics is a constantly expanding branch of number theory, with more and more people being drawn into the complex subtleties of Fibonacci's legacy.

Here are two interesting Properties of Fibonacci sequence:

The Mystery of the three Consecutive Numbers in the Fibonacci sequence

In the Fibonacci sequence, if you take any three consecutive numbers, add them, and divide the sum by 2, you always get the third number.

Let’s take a Fibonacci sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 ….

1+2+3=6 and 6/2=3

5+8+13=26 and 26/2=13

8+13+21=42 and 42/2=21 

21+34+55=110 and 110/2=55

89+144+233=466 and 466/2=233

377+610+987=1974 and 1974/2=987

…and it goes on to infinity. 

The Mystery of the Four Consecutive Numbers in the Fibonacci sequence

Take any four consecutive numbers in the sequence other than ‘0’. Multiply the outer numbers, then multiply the inner numbers. Subtract them. The difference is 1.

Example 1

0, 1, 1, 2, 3, 5, 8, 13, 21, 34…

8(2) – 5(3) = 16 – 15 = 1

Example 2

1, 1, 2, 3, 5, 8, 13, 21, 34, 55…

21(5) – 8(13) = 105 – 104 = 1

Example 3

1, 2, 3, 5, 8, 13, 21, 34, 55, 89…

34(8) – 21(13) = 272 – 273 = -1