math15

Fibonacci
Sequence

What is Fibonacci Sequence?

The Fibonacci sequence is a type series where each number is the
sum of the two that precede it. It starts from 0 and 1 usually. The
Fibonacci sequence is given by 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
144, and so on. The numbers in the Fibonacci sequence are also
called Fibonacci numbers. In Math, the sequence is defined as an
ordered list of numbers that follow a specific pattern. The numbers
present in the sequence are called the terms. The different types of
sequences are arithmetic sequence, geometric sequence, harmonic
sequence and Finbonacci Sequence.

What is Fibonacci Sequence?

Fibonacci Sequence = 0, 1, 1, 2, 3, 5, 8, 13, 21, ….

Here, the third term “1” is obtained by adding the first and second
term. (i.e., 0+1 = 1)
Similarly,
“2” is obtained by adding the second and third term (1+1 = 2)
“3” is obtained by adding the third and fourth term (1+2) and so on.
For example, the next term after 21 can be found by adding 13 and
21. Therefore, the next term in the sequence is 34.

What is Fibonacci Sequence?

The Fibonacci sequence of numbers “𝐹𝑛” is defined using the
recursive relation with the seed values 𝐹0 = 0 and 𝐹1 = 1 :

𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2

Here, the sequence is defined using two different parts, such as

kick-off and recursive relation.

The kick-off part is 𝐹0 = 0 and 𝐹1 = 1 .
The recursive relation part is 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2.
It is noted that the sequence starts with 0 rather than 1.
So, 𝐹5 should be the 6th term of the sequence.

Find me!

Find the 1st 20 term in a Fibonacci Sequence.

0, 1, 1, 2,……………………………………….

Golden Ratio

The Fibonacci Sequence is closely related to the value of the
Golden

Ratio.

We

know

that

the

Golden

Ratio

value

is

approximately equal to 1.618034. It is denoted by the symbol “φ”. If
we take the ratio of two successive Fibonacci numbers, the ratio is
close to the Golden ratio. For example, 3 and 5 are the two
successive Fibonacci numbers. The ratio of 5 and 3 is:
5
3= 1.6666

Take another pair of numbers, say 21 and 34, the ratio of 34 and 21
is:
34
21= 1.619

Golden Ratio

It means that if the pair of Fibonacci numbers are of bigger value,
then the ratio is very close to the Golden Ratio.
So, with the help of Golden Ratio, we can find the Fibonacci
numbers in the sequence.
The formula to calculate the Fibonacci numbers using the Golden
Ratio is:

𝑥𝑛 =
𝜑𝑛− 1 − 𝜑𝑛

5

Where,
φ is the Golden Ratio, which is approximately equal to the value of

1.618

n is the nth term of the Fibonacci sequence.

Let us check!

Guide Questions:
1. What is Fibonacci Sequence?
2. Why Fibonacci Sequence is significant?
3. What are two different ways to find the Fibonacci Sequence?
4. What is the value of the Golden ratio?

Mathematics

For our World

Mathematics for our World

❖Mathematics for Organization

❖Mathematics for Prediction

❖Mathematics for Control

Summary

❖Patterns and Mathematics in Nature

▪

Patterns

are

regular,

repeated,

reoccurring

forms

or

designs. Patterns are commonly observed in natural objects
such as the six-fold symmetry or snowflakes, the hexagonal
structure and formation of honeycombs, the tiger’s stripes
and hyena’s spot, the number of seeds in a sunflower, the
spiral snail’s shell and the number of petals of flowers.

Summary

❖The Fibonacci Sequence

▪

Sequence is an ordered list of numbers, called terms, that
may have repeated values. The arrangement of these terms
is set by a definite rule. The terms of a sequence could be
generated by applying the rule to previous terms of the
sequence.

▪

The Fibonacci sequence is the sequence of numbers, in
which every term in the sequence is the sum of terms
before it, beginning with 0 and 1. Ratios of two Fibonacci
numbers

approximate

the

Golden

Ration,

which

is

considered as the most aesthetically pleasing proportion.

Summary

❖Mathematics for our World

▪

Mathematics helps organize patterns and irregularities in
the world. Mathematics helps predict the behavior of
nature and phenomena in the world, as well as helps
humans exert control over occurrences in the world for the
advancement of our civilization.

Exit Ticket

How is it possible that Mathematics, a
product

of

human

thought

that

is

independent

of

experience,

fits

so

excellently the objects of reality?

-Albert Einstein-