Education

# Fibonacci Sequence – Introduction and Definition

The sequence is a mathematical term that refers to an ordered series of positive integers that follow a specified pattern. Different types of sequences are arithmetic, geometric, harmonic, and Fibonacci. The Fibonacci sequence is a set of numbers in which every next number is the sum of the two preceding numbers. Beginning with ”zero” and “one”, the sequence is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.
The Fibonacci sequence is a pattern-following series of infinite numbers. By combining the two preceding numbers in the sequence together, the next number in the sequence can be discovered.

## Fn = Fn-1 + Fn-2, where n represents a number in the series and F represents the Fibonacci number value.

The Fibonacci Sequence is a set of numbers that begins with 0 and ends with 1, with each number equal to the sum of the two numbers before it.

The Fibonacci Sequence is composed of the numbers 0, 1, 1, 2, 3, 5, 8, 13, 21,34, and so on. “1” is the third term, and we get 1 by adding the first and second terms. (In other words, 0+1 Equals 1) Similarly, we get 2 (1+1 = 2) by adding the 2nd and 3rd terms. We get 3 (1+2) by adding the third and fourth terms, and so on. For example, combining 21 and 34 yields the next term ie 55

## Fibonacci Sequence Formula

The Fibonacci numbers form a sequence when each number is the sum of the two before it. ‘0’ and ‘1’ are the first two. Mathematically the formula for the Fibonacci series is given:
Xn= Xn-1 + Xn-2 where n is the number in the series
So the third number will be 0 + 1 =1
The fourth number will be 1 +1 =2
The fifth number will be 2 + 1= 3
The sixth number will be 3 +2=5
And so on.

## Examples

Example 1: Find the Fibonacci number when n=6, using recursive relation.
Solution:
The formula to calculate the Fibonacci sequence is: Xn = Xn-1+Xn-2
Take: F0=0 and F1=1
Using the formula, we get
X2 = X1+X0 = 1+0 = 1

X3 = X2+X1 = 1+1 = 2

X4 = X3+X2 = 2+1 = 3

X5 = X4+X3 = 3+2 = 5

X6=X5 + X4 = 8

Therefore, the Fibonacci number is 8.

## Uses of Fibonacci Sequence

• ### The Fibonacci sequence can be easily found in nature.

Many natural things follow the Fibonacci sequence, as we can see. It can be found in biological settings such as tree branching, the arrangement of leaves on a stem, pineapple fruit sprouts, artichoke flowering, unfurling ferns, and the arrangement of pine cone bracts, among others.

• ### The Fibonacci sequence is employed in engineering.

The Fibonacci sequence is used in computer data structures and sorting algorithms, financial engineering, audio compression, and architectural engineering. The Fibonacci sequence can be found in nature in the spirals of sunflower seeds and the form of a snail’s shell.

## Fibonacci Numbers

Basically, Fibonacci numbers are a series of numbers derived from adding the two numbers preceding it. In other words, the next number in a series is the result of adding two previous numbers.

For instance, let’s use 0 and 1 for the first two integers in the series. We get the third number as 1 when we add 0 and 1. The fourth number is obtained by adding the second and third numbers (i.e. 1 and 1), and the process continues in this manner. As a result, the Fibonacci sequence becomes 0, 1, 1, 2, 3, 5, 8,13, 21,34,…… As a result, the series obtained is known as the Fibonacci number series.