The document discusses the Fibonacci sequence and some of its properties. It begins by defining the Fibonacci sequence as a series of numbers where each number is the sum of the two preceding ones, starting with 0 and 1. It then explains how the golden ratio is represented by the Greek letter φ and relates to the ratio of consecutive Fibonacci numbers. Finally, it derives a general formula for determining the nth term in the Fibonacci sequence.

1. FIBONACCI SERIES
BY-
KULDEEP GUPTA

2. CONTENTS
INTRODUCTION
FUN WITH FIBONACCI NUMBERS
THE GOLDEN RATIO
FIBONACCI NUMBERS IN NATURE
GENERAL TERM FOR A FIBONACCI NUMBER
SUMMARY
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3. INTRODUCTION
The Fibonacci numbers are the
numbers in the following integer
sequence, called the Fibonacci
sequence, and characterized by the
fact that every number after the first
two is the sum of the two preceding
ones:
0,1,1,2,3,5,8,13,21,34,55,89,144,….
LEONARDO PASINO
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4. FUN WITH FIBONACCI NUMBERS
1 1 2 3 5 8 13 21 34 55 89 …
1 1 4 9 25 64 169 441 1156 3025 7921 …
12+12+22=6=2x3
12+12+22+32=15=3x5
12+12+22+32+52+82 =104=8x13
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5. THE GOLDEN RECTANGLE
12
22
32
52
82
12
5

6. THE GOLDEN RATIO
Ratio of two consecutive Fibonacci
numbers.
Represented by Greek letter φ(phi).
Φ=
1+ 5
2
=1.618033988749894848204586834
1.49
1.5
1.51
1.52
1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.6
1.61
1.62
1.63
1.64
1.65
1.66
1.67
1.68
0
Φ v/s F
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7. FIBONACCI NUMBERS IN NATURE
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8. GENERAL TERM FOR A FIBONACCI NUMBER
General representation of a Fibonacci sequence :
Fn=Fn-1+Fn-2
Consider a sequence
f(y)= 𝑛=0
∞
𝐹 𝑛 𝑦 𝑛
where F is the Fibonacci numbers starting from 0 i.e. F0=0 and so on
Hence,
f(y)=1y1+1y2+2y3+3y4+………………………..
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9. GENERAL TERM FOR A FIBONACCI NUMBER(contd.)
f(y)=1y1+1y2+2y3+3y4+……………………………+Fnyn +………
yf(y)= 1y2+1y3+2y4+3y5+5y6+…………..…+Fn-1yn-1+............
y2f(y)= 1y3+1y4+2y5+3y6+5y7+………+Fn-2yn-2+…………….. Subtracting
the sum of second and third from first,
f(y)-yf(y)-y2f(y)=y
Hence,
f(y)=
𝑦
1−𝑦−𝑦2
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10. GENERAL TERM FOR A FIBONACCI NUMBER(contd.)
f(y)=
𝑦
1−𝑦−𝑦2
Φ=
1+ 5
2
Using partial fractions in f(y) and then incorporating value of Φ, we get
f(y)=
1
5
(
1
1−(yΦ)
−
1
1−𝑦(1−Φ)
)
Taking y such that |y Φ|<1 and |y(1- Φ)|<1,we can apply power series
expansion-
1
1−𝑥
= 𝑛=0
∞
𝑥 𝑛
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11. GENERAL TERM FOR A FIBONACCI NUMBER(contd.)
Therefore,
f(y)=
1
5
( 𝑛=0
∞
(yΦ) 𝑛- 𝑛=0
∞
(𝑦(1 − Φ)) 𝑛
f(y)=
1
5
( 𝑛=0
∞
((Φ) 𝑛-(1 − Φ)n)yn
Comparing it with,
f(y)= 𝑛=0
∞
𝐹 𝑛 𝑦 𝑛
We get,
𝐹 𝑛=
1
5
(Φn-(1- Φ)n)
Where, Φ =
1+ 5
2
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12. SUMMARY
Fibonacci Sequence has been around for hundreds of years.
Golden Ratio has been used for thousands of years.
Both concepts are still in use today and continue to interest us.
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13. REFERENCES
https://en.wikipedia.org/wiki/Fibonacci_number
https://www.youtube.com/watch?v=CR-nmp97Ayo
 https://www.livescience.com/37470-fibonacci-sequence.html
https://www.youtube.com/watch?v=nt2OlMAJj6o
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