By:- Arya Pratap
XI A
Fibonacci
Sequence
WHAT IS FIBONACCI
SEQUENCE?
 The Fibonacci series is the sequence of
numbers (also called Fibonacci numbers),
where every number is the sum of the
preceding two numbers, such that the first
two terms are '0' and '1'.
 In some older versions of the series, the
term '0' might be omitted. A Fibonacci
series can thus be given as, 0, 1, 1, 2, 3, 5, 8,
13, 21, 34, . . . It can be thus be observed
that every term can be calculated by adding
the two terms before it.
WHAT IS FIBONACCI
SEQUENCE?
 Given the first term, F0 and second term,
F1 as '0' and '1', the third term here can be
given as, F2 = 0 + 1 = 1
Similarly,
F3 = 1 + 1 = 2
F4 = 2 + 1 = 3
 Therefore, to represent any (n+1)th term in
this series, we can give the expression as,
Fn = Fn-1 + Fn-2. We can thus represent a
Fibonacci series as shown in the image
below,
Discovery

The Fibonacci numbers were first discovered by a man
named Leonardo Pisano (The Italian mathematician, who
was born around A.D. 1170, was initially known as
Leonardo of Pisa.).

The Fibonacci sequence can elaborately written as
{1,1,2,3,5,8,13,21,34,55,89,144,233…….}.

One of the most common experiments dealing with the
Fibonacci sequence is his experiment with rabbits.

Fibonacci put one male and one female rabbit in a field.
Fibonacci supposed that the rabbits lived infinitely and
every month a new pair of one male and one female
was produced.

Fibonacci asked how many would be formed in a year.
Following the Fibonacci sequence perfectly the rabbits
reproduction was determined...144 rabbits.

Though unrealistic, the rabbit sequence allows people to
attach a highly evolved series of complex numbers to an
everyday, logical, comprehendible thought.

Bortner and Peterson (2016) elaborately described the
history and application of Fibonacci numbers.
Discovery

Indian mathematics as far back as 200 BC, the
Fibonacci Sequence eventually got its name from
the Italian mathematician Leonardo of Pisa — a.k.a.
Fibonacci — who detailed the formula in his book
Liber Abaci (1202).

In his book, the Fibonacci Sequence was used for
describing the growth pattern of the rabbit
population, where the sum of the formula was used
for hypothesizing about a rabbit’s breeding pattern.
What’s so fascinating about this concept is that the
formula often appears out of the blue in
mathematics, often unexpectedly and often
without trying to find it in the first place. It even
appears in nature, such as in the pattern of
branching in trees or the placement of
a stem’s leaves.
Fibonacci Series
Formula
• The Fibonacci series formula in mathematics
can be used to find the missing terms in a
Fibonacci series. The formula to find the
(n+1)th term in the sequence is defined using
the recursive formula, such that F0 = 0, F1 =
1 to give Fn.
• The Fibonacci formula is given as follows.
Fn = Fn-1 + Fn-2, where n > 1
FIBONACCI SERIES
SPIRAL
• The Fibonacci spiral is the representation of the
pattern formed by the Fibonacci numbers in a
grid. The Fibonacci series spiral starts in a plane
in the shape of a rectangle whose dimensions
(length × breadth) follow the principle of a
"Golden Ratio" (≈1.618), and is therefore referred
to as "Golden Rectangle". The following image
depicts the Fibonacci spiral starting with a
rectangle partitioned into 2 squares. Fibonacci
spiral is an approximation of the golden spiral.
Fibonacci
Series List
 Each term of a Fibonacci series is a sum of
the two terms preceding it, given that the
series starts from '0' and '1'. We can use
this to find the terms in the series. The first
20 numbers in a Fibonacci series are given
below in the Fibonacci series list.
F0 = 0
F10 = 55
F1 = 1
F11 = 89
F2 = 1
F12 = 144
F3 = 2
F13 = 233
F4 = 3
F14 = 377
F5 = 5
F15 = 610
F6 = 8
F16 = 987
F7 = 13
F17 = 1597
F8 = 21
F18 = 2584
F9 = 34
F19 = 4181
THE GOLDEN RATIO
• Any term in the Fibonacci sequence divided by the
previous has a quotient of approximately 1.618034….
That is, an /an-1≈1.618034. For the first few terms,
this is a very loose approximation, but as the term
number (n) increases, the quotient coincides more
exactly with this irrational value. The ratio between 1
and 1.618034 is known as the Golden Ratio
(abbreviated 𝛗𝛗), and a rectangle with a width to
height ratio of 1:1.618034 is known as the Golden
Rectangle.
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THE END