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Name:- Rishinandan
Madhavan
Class:- 12th B
Roll no.:- 23
School:- Ryan International
School, Sanpada
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CERTIFICATE
This is to certify that Rishinandan Madhavan, student of Class
12th B, Ryan International School, Sanpada, has successfully
completed the coursework of Mathematics on the topic
“Fibonacci Sequence, it’s properties and applications“, under
the guidance of Ms. Sonia Saxena during the academic
session 2023 – 2024 in partial fulfilment of Mathematics
Practical Examination of the Central Board of Secondary
Education (CBSE).
External Signature:
Principal’s Signature:
Internal Signature:
School Stamp:
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ACKNOWLEDGEMENT
I would like to express my sincere and special gratitude
to our Principal Ma’am, Mrs. Muriel Fernandes of our
esteemed Ryan International School, Sanpada for
always encouraging us to excel in all that we do. I
would like to thank my Mathematics teacher, Ms. Sonia
Saxena for her continuous guidance and encouragement
and immense motivation which helped me at all stages
of this project. Lastly, I would like to thank my family
and friends for helping me in the completion of the
project.
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INDEX
Topic Page Number
Introduction 6
History of the 8
Fibonacci sequence
Golden ratio 10
Relation to the 12
golden ratio
Golden ratio in 14
nature
Applications of 16
Fibonacci sequence
Bibliography 17
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Introduction
In mathematics, the Fibonacci sequence is a
sequence in which each number is the sum of
the two preceding ones. Numbers that are part
of the Fibonacci sequence are known as
Fibonacci numbers, commonly denoted Fn . The
sequence commonly starts from 0 and 1,
although some authors start the sequence from
1 and 1 or sometimes (as did Fibonacci) from 1
and 2. Starting from 0 and 1, the first few values
of the sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,
55, 89, 144.
Fibonacci numbers appear unexpectedly often
in mathematics, so much so that there is an
entire journal dedicated to their study, the
Fibonacci Quarterly.
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Applications of the Fibonacci numbers include
computer algorithms such as the Fibonacci
search technique, and the Fibonacci heap data
structure, and graphs called Fibonacci cubes
used for interconnecting parallel and
distributed systems.
They also appear in biological settings, such as
branching in trees, the arrangement of leaves
on a stem, the fruit sprouts of a pineapple, the
flowering of an artichoke, an uncurling fern,
and the arrangement of a pine cone’s bracts.
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History of the Fibonacci
sequence
In Europe:-
The Fibonacci sequence first appears in the
book Liber Abaci (The Book of Calculation,
1202) by Fibonacci where it is used to calculate
the growth of rabbit populations. Fibonacci
considers the growth of an idealized
(biologically unrealistic) rabbit population
assuming that: a newly born breeding pair of
rabbits are put in a field; each breeding pair
mates at the age of one month and at the end of
their second month they always produce
another pair of rabbits; and rabbits never die,
but continue breeding forever. Fibonacci posed
the puzzle: how many pairs will there be in one
year?
At the end of the first month, they mate, but
there is still 1 pair.
At the end of the second month they produce a
new pair so there are 2 pairs in the field.
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At the end of the third month, the original pair
produce a second pair, but the second pair only
mate to gestate for a month, so there are 3 pairs
in all.
At the end of the fourth month, the original pair
has produced yet another new pair, and the pair
born two months ago also produces their first
pair making 5 pairs.
At the end of the nth month, the number of
pairs of rabbits is equal to the number of
mature pairs (that is, the number of pairs in
month n – 2) plus the number of pairs alive
last month (month n – 1). The number in the
nth month is the nth Fibonacci number.
The name “Fibonacci sequence was first used
in the 19th-centure number theorist Édouard
Lucas.
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Golden ratio
In mathematics, two quantities are in the
golden ratio if their ratio is the same as the
ratio of their sum to the larger of the two
quantities. Expressed algebraically, for
quantities a and b with a > b > 0.
a+b a
= =φ
a b
Where the Green letter phi (φ∨∅ ) denotes the
golden ratio. The constant φ satisfies the
quadratic equation φ =φ+1 and is an irrational
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number with a value of
1+ √ 5
φ= =1.618 033 988 749 … .
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The golden ratio was called the extreme and
mean ratio by Euclid, and the divine
proportion by Luca Pacioli, and also goes by
several other names.
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Mathematicians have studied the golden
ratio’s properties since antiquity. It is the
ratio of a regular pentagon’s diagonal to its
side and thus appears in the construction of
the dodecahedron and icosahedron. A golden
rectangle – that is, a rectangle with an aspect
ratio of φ may be cut into a square and a
smaller rectangle with the same aspect ratio.
The golden ratio has been used to analyse the
proportions of natural objects and artificial
systems such as financial markets, in some
cases based on dubious fits to data. The
golden ratio appears in some patterns in
nature, including the spiral arrangement of
leaves and other parts of vegetation.
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Relation to the golden
ratio
Like every sequence defined by a linear
recurrence with constant coefficients, the
Fibonacci numbers have a closed-form
expression. It has become known as Binet’s
formula, named after French mathematician
Jacques Philippe Marie Binet, though it was
already known by Abraham de Moivre and
Daniel Bernoulli:
n n n n
φ −ψ φ −ψ
F n= = ,
φ−ψ √5
where
1+ √5
φ= ≈−0.6180339887 … .
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since ψ=−φ ,
−1
this formula can also be written as
n −n n −n
φ −(−φ) φ −(−φ)
F n= = .
√5 2 φ−1
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To see the relation between the sequence and
these constants, not that φ and ψ are both
solutions of the equation x2 = x + 1 and thus
xn = xn-1 + xn-2, so the powers of φ and ψ satisfy
the Fibonacci recursion. In other words,
n−1 n−2
φ=φ +φ
n −1 n−2
ψ=ψ +ψ
It follows that for any values a and b, the
sequence defined by
n n
U n =a φ + bψ
Satisfies the same reference,
n n
U n =a φ + bψ
U n =a ( φ n−1+ φn−2 ) +b ( ψ n−1 +ψ n−2 )
n −1 n−1 n−2 n−2
U n =a φ +b ψ +a φ +bψ
U n =U n−1+ U n−2
If a and b are chosen so that U0 = 0 and U1 = 1
then the resulting sequence must be the
Fibonacci sequence.
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Golden ratio in nature
1. Seed Heads:
2. Tree Branches:
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3. Shells:
4. Spiral Galaxies:
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Applications of Fibonacci
sequence
The Fibonacci sequence can be found in a
varied number of fields from nature, to music,
and to the human body.
used in the grouping of numbers and the brilliant
proportion in music generally.
used in Coding (computer algorithms, interconnecting
parallel, and distributed systems)
in numerous fields of science including high-energy
physical science, quantum mechanics, Cryptography,
etc.
used to model various phenomena in biology, such as
the growth patterns of plants and the arrangement of
leaves on a stem.
used in financial analysis to identify trends in stock
prices and other financial data.
You can use the Fibonacci calculator that helps
to calculate the Fibonacci Sequence. Look at a
few solved examples to understand the
Fibonacci formula better.
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Bibliography
1. Fibonacci sequence – Wikipedia:
https://en.wikipedia.org/wiki/
Fibonacci_sequence
2. Golden ratio – Wikipedia:
https://en.wikipedia.org/wiki/Golden_ratio
3. Golden ratio in nature – Mathnasium:
https://www.mathnasium.com/blog/14-
interesting-examples-of-the-golden-ratio-in-
nature#:~:text=The%20Fibonacci
%20sequence%20can%20also,each%20of
%20the%20new%20stems.
4. Applications of Fibonacci sequence –
Cuemath:
https://www.cuemath.com/numbers/
fibonacci-sequence/
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