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
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