About Fibonacci Numbers..

1. Fibonacci
Numbers
Nithishwaran.S
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2. What is Fibonacci Number?
A Fibonacci number is a series of numbers in which each
Fibonacci number is obtained by adding the two preceding numbers.
It means that the next number in the series is the addition of two
previous numbers. Let the first two numbers in the series be taken as
0 and 1. By adding 0 and 1, we get the third number as 1. Then by
adding the second and the third number (i.e.) 1 and 1, we get the
fourth number as 2, and similarly, the process goes on. Thus, we get
the Fibonacci series as 0, 1, 1, 2, 3, 5, 8, ……. Hence, the obtained
series is called the Fibonacci number series.
Fibonacci

3. We can also obtain
the Fibonacci
numbers from the
Pascal's triangle as
shown in the figure
shown in left.
Here, the sum of
diagonal elements
represents
the Fibonacci
sequence, denoted by
color lines.

4. Fibonacci Series List
The list of numbers of Fibonacci Sequence is given
below. This list is formed by using the formula, which is
mentioned in the above definition.

5. Fibonacci Numbers Formula
– The sequence of Fibonacci numbers can be defined as:
– Fn = Fn-1 + Fn-2
– Where Fn is the nth term or number
– Fn-1 is the (n-1) th term
– Fn-2 is the (n-2) th term

6. Fibonacci Numbers in Using
formula
Fn-1 Fn-2 Fn
0 1 1
1 1 2
1 2 3
2 3 5
3 5 8
5 8 13
8 13 21
13 21 34
21 34 55
34 55 89

8. Fibonacci Number Properties
– The following are the properties of the Fibonacci numbers.
– In the Fibonacci series, take any three consecutive numbers and add those
numbers. When you divide the result by 2, you will get the three numbers. For
example, take 3 consecutive numbers such as 1, 2, 3. when you add these
numbers, i.e. 1+ 2+ 3 = 6. When 6 is divided by 2, the result is 3, which is 3.
– Take four consecutive numbers other than “0” in the Fibonacci series. Multiply
the outer number and also multiply the inner number. When you subtract these
numbers, you will get the difference “1”. For example, take 4 consecutive
numbers such as 2, 3, 5, 8. Multiply the outer numbers, i.e. 2(8) and multiply
the inner number, i.e. 3(5). Now subtract these two numbers, i.e. 16-15 =1.
Thus, the difference is 1.

10. The Rabbits of Fibonacci
and the famous sequence
Liber Abaci, in addition to referring to Indo-
Arabic numbers, which subsequently took the
place Roman numerals, also included a large
collection of problems addressed to merchants,
concerning product prices, calculation of business
profit, currency conversion into the various coins
in use in the Mediterranean states, as well as
other problems of Chinese origin. Alongside these
commercial problems were others, much more
famous, which also had a great influence on later
authors. Among them, the most famous, source of
inspiration for many mathematicians of later
centuries, is the following:
“How many pairs of rabbits will be born in a year, starting from a single pair, if each month each pair gives birth to a
new pair which becomes reproductive from the second month?”. The solution to this problem is the famous
“Fibonacci sequence”: 0, 1, 1, 2, 3, 5, 8, 13, 21,34,55,89… a sequence of numbers in which each member is the sum
of the previous two.

11. Examples of Fibonacci Sequences in
Our Life are as follows

12. The Fibonacci as some
of the largest structures
in the universe. Spiral
galaxies are the most
common galaxy shape.
Galaxies group together
in super clusters and
super clusters group
together in walls.
Galaxies

13. Tree branches
The Fibonacci sequence
can also be seen in the way
tree branches form or split. A
main trunk will grow until it
produces a branch, which
creates two growth points.
Then, one of the new stems
branches into two, while the
other one lies dormant. This
pattern of branching is
repeated for each of the new
stems. A good example is the
sneezewort.

14. Africa
One blogger has
applied the Fibonacci
sequence to
population density and
land mass. In Africa
the majority of highly
populated cities fall on
or close to where the
spiral predicts

15. Snails and
fingerprints.

16. Katsushika
Hokusai
The Great Wave at
Kanagawa (from a Series
of Thirty–Six Views of
Mount Fuji), Edo period
(1615–1868), ca. 1831–33.

17. Water falls into
the shapes of a
Fibonacci during
numerous
events. Another
example would
be a vortex.

18. Fibonacci in spores. A fiddlehead or koru. A monarch caterpillar about to
form a chrysalis.
.

19. Aloe Plant
Spiral aloe. Numerous
cactus display the
Fibonacci spiral. You
can see how each set
of leaves spiral
outward.

20. Double
Fibonacci
This flower exhibits
two Fibonacci
spirals. You can
faintly see how the
spirals form from the
center of the
opened disk florets.

21. Chameleon Tail
The tail of these
creatures naturally
curls into a
Fibonacci spiral.

22. Pinecones
When looking closely at the
seed pod of a pinecone,
you’ll notice an arranged
spiral pattern. Each cone
has its own set of spirals
moving outwards in
opposing directions.

23. Shell Fossil
A Shell Fossil with the
Fibonacci sequence.
You can see as the
shell grew, a Fibonacci
spiral was formed.

24. Romanesque
cauliflower
Take a good look at
this Romanesque
cauliflower -- its
spiral follows the
Fibonacci sequence.

25. You!
Yes! You are an example of
the beauty of the Fibonacci
Sequence. The human body
has various representations
of the Fibonacci Sequence
proportions, from your face
to your ear to your hands
and beyond!
You have now been
proven to be mathematically
gorgeous, so go forth and be
beautiful! ...and maybe think
math is a little bit better than
you first thought?

26. Life is a math equation. In order to
gain the most, you have to know how
to convert negatives into positives.
Thank You!
Have A Good Day To all