Fibonacci Sequence

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What is the Fibonacci Sequence?

The Fibonacci sequence, which is also known as Fibonacci numbers, is defined as the sequence of numbers in which each number in the sequence is equal to the sum of two numbers that occur before it.

The Fibonacci Sequence in Mathematics is given as:

The numbers in Fibonacci Sequence are as follows:  0, 1, 1, 2, 3, 5, 8, 13, 21, ….

Here, the third term “1” is obtained by adding the first and second terms. (i.e., 0+1 = 1)

Similarly,

The term “2” is obtained by adding the second and third term (1 + 1 = 2)

The term “3” is obtained by adding the third and fourth term (1 + 2) and so on.

For example, the next term after the number 21 can be found by adding 13 and 21. Therefore, the next term we get in the sequence is 34.


Fibonacci Sequence Formula

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

Where, Fn equals  Fn-1 + Fn-2

Here, the Fibonacci sequence is defined using two different parts, such as the kick-off relation and recursive relation.

The kick-off part is F0 equals 0 and F1 equals 1.

The recursive relation part is the term Fn which equals Fn-1 + Fn-2.

It is noted that the sequence starts with the number 0 rather than the number 1.

So, F5 should be the 6th term of the given Fibonacci sequence.


Fibonacci Sequence List

The list of first 20 terms in the Fibonacci Sequence is given below:


0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, and the number 4181.


The list of Fibonacci numbers are calculated as follows:


Fn

List of Fibonacci Numbers

0

0

1

1

2

1

3

2

4

3

5

5

6

8

7

13

8

21

9

34

… and so on.

… and so on.


Why is Fibonacci in Nature?

In nature, the growth, as well as the self-renewal of cell populations, lead to the generation of hierarchical patterns in tissues that resemble the pattern of population growth in rabbits, which is explained by the classic Fibonacci sequence.

The first term is as a1 equals 1 and a2 equals 1. Now from the third term onwards, each and every term of this Fibonacci sequence will become the sum of the previous two terms. So a3 will be given as the sum of a1 as well as the term a2.

Therefore, 1 + 1 equals 2. Similarly,

a4 equals a2 + a3

∴ 1 + 2 equals 3

a5 equals a3 + a4

∴ 2 + 3 equals 5

Therefore if we want to write the Fibonacci sequence, we can write it as, [1  1  2  3  5…and so on]. So, in general, we can say,

an equals an-1 + an-2

where the value of the variable n ≥ 3.


Types of Sequences

  • Arithmetic Sequence - In arithmetic or linear sequence, the difference between any two consecutive terms is constant.

  • Quadratic Sequence - A quadratic sequence can be defined as a sequence of numbers in which the second difference between any two consecutive terms is constant.

  • Geometric Sequence - We can define a geometric sequence as a sequence of numbers where each term after the first term is found by multiplying the previous one by a fixed, non-zero number which is known as the common ratio.

The Fibonacci Spiral and The Golden Ratio

The Fibonacci sequence can be often visualized in a graph. Each of the squares illustrates the area of the next number in the Fibonacci sequence. The Fibonacci spiral is then drawn inside the squares easily by connecting the corners of the boxes.


(Image will be uploaded soon)


The squares fit together perfectly and this is because the ratio between the numbers in the Fibonacci sequence is very close to the golden ratio [1], which is approximately equal to 1.618034. The larger the numbers in the Fibonacci sequence, it concludes that the closer the ratio is to the golden ratio.

The spiral, as well as the resulting rectangle, are also known as the Golden Rectangle [2].


The Origins of The Fibonacci Sequence

Fibonacci (Leanardo Pisano Bogollo, Fibonacci was his nickname) first introduced the series of numbers known as the Fibonacci sequence in his book Liber Abaci in 1202. Fibonacci was a member of an important Italian trading family in the 12th and 13th centuries. Being part of a trading family, mathematics was an important part of the business. Fibonacci travelled throughout the Middle East as well as India and was captivated by the mathematical ideas from his travels. His book, Liber Abaci, was a discourse on the mathematical methods in commerce that Fibonacci observed during his travels to different parts.


We remember Fibonacci for two important contributions to Western mathematics:

  1. He was the one who helped spread the use of Hindu systems of writing numbers in Europe (0, 1, 2, 3, 4, 5 in place of the Roman numerals).

  2. The seemingly insignificant series of the numbers later named the Fibonacci Sequence after Fibonacci.

FAQ (Frequently Asked Questions)

Question 1. What are the First Ten Fibonacci Numbers?

Answer. So the term F10 is really the 11th element of the Fibonacci sequence. The first twenty elements of the Fibonacci Sequence are as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, and 4181.

Question 2. What is the Use of the Fibonacci Series?

Answer. The use of Fibonacci numbers, numbers in the Fibonacci sequence can be used to create technical indicators using a mathematical sequence developed by the Italian mathematician, commonly referred to as the "Fibonacci," in the 13th century. The sequence of numbers, starting with the numbers zero as well as one is created by adding the previous two terms.

Question 3. Is the First Fibonacci Number 0 or 1?

Answer. By definition, the first two Fibonacci numbers are the terms 0 and 1, and each remaining number is the sum of the previous two numbers. Some sources omit the initial 0, instead of beginning the sequence with two number of 1s.

Question 4. What is Fibonacci in Nature?

Answer. In nature, the growth, as well as self-renewal of cell populations, leads to the generation of hierarchical patterns in tissues that resemble the pattern of population growth in rabbits, which is explained by the classic Fibonacci sequence.