The document discusses the Fibonacci sequence, a series created by Leonardo Fibonacci where each number is the sum of the two preceding ones, and its close relation to the golden ratio (approximately 1.618). It details the mathematical formulation of the sequence, how the ratio of successive Fibonacci numbers approaches the golden ratio, and various applications of the golden ratio in fields like architecture and nature. Interestingly, it highlights that Fibonacci numbers often appear in the natural world and concludes with the celebration of Fibonacci Day on November 23.

1. Fibonacci Sequence and
Golden Ratio
-Aditya Garg
11-C
Vasant Valley School

2. Contents
 Fibonacci sequence
 How does Fibonacci sequence work
 Golden Ratio
 Applications of Golden Ratio
 Some interesting facts

3. Fibonacci sequence
 It is series of numbers that follow a unique integer sequence.
 Inventedby Leonardo Fibonacci.
 It is closely related to Lucas numbers and Golden Ratio.
 The Fibonacci numbers are 1,1,2,3,5,8,13,etc.

4. How does the Fibonacci sequence
work
 The numbers of the series are obtained by adding the two previous
numbers in the sequence to obtain the next higher number.
 Example: 1+1=2,1+2=3,3+2=5,5+8=13,etc.
 Formulais: An=An-1+An-2 for n>=3 where :
An is term number "n“
An-1 is the previous term (n-1)
An-2 is the term before that (n-2)
A1=A2=1
 When we take any two successive (one after the other) Fibonacci
Numbers, their ratio is very close to the GOLDEN RATIO (1.618034).

5. Golden Ratio
 The golden ratio (symbol is the Greek letter "phi(φ)) is a special number
approximately equal to 1.618
 It can also be calculates as 2 x Sin (54 degree)
 It appears many times in geometry, art, architecture and other areas.
 The ratio of each successive pair of numbers in the Fibonacci Sequence
converge on the golden ratio as we go higher in the sequence.
3/2 = 1.5
5/3 = 1.66666666666666…
8/5 = 1.6
3/8 = 1.625
21/13 = 1.61538461538462…
34/21 = 1.61904761904762…
55/34 = 1.61764705882353…
89/55 = 1.61818181818182…

6. Applications Of Golden Ratio
Architecture
Painting
Book Design
Nature
Aesthetics
Perceptual Studies

7. Do you know?
 Fibonacci numbers frequently appear in the numbers of petals in
a flower and in the spirals of plants.
 The positions and proportions of the key dimensions of many animals are
based on Phi(φ).
 The diameters of the Earth and Moon form a triangle whose dimensions
are based on the mathematical characteristics of phi. The distances of
the planets from the sun correlate surprisingly closelyto exponential
powers of Phi(φ).
 Golden Ratio is used in the design of the Apple logo!!!

8. Golden Ratio is Everywhere!!!

9. Fibonacci Day?
 NOVEMBER 23. WHY?
 11/23
 1 1 2 3

10. THANK YOU