Fibonacci Sequence

Fibonacci Sequence was discovered by Italian Mathematician Leonardo Fibonacci (1170-1250) is a sequence of numbers starting with zero and one , is a steadily increasing series where each number is equal to the sum of the preceding two numbers.

Definition

Fibonacci Sequence is defined as the sequence of numbers in which each number in the sequence is equal to the sum of two numbers before it.

Fibonacci sequence is given as $0,1,1,2,3,5,8,13.........$

Here the first two terms are $0,1$

The third term $1$ is obtained by adding the second and first term $(1+0=1)$

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

The fifth term $3$ is obtained by adding the fourth and third term $(2+1=3)$ and so on.

Fibonacci Sequence Formula

The Fibonacci sequence of numbers $F_{n}$ is defined using the recursive relation with the seed values $F_{0}=0$ and $F_{1}=1$.

$F_{n}=F_{n-1}+F_{n-2}$

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

$F_{0}=0$ and $F_{1}=1$ is the the kick-off part

$F_{n}=F_{n-1}+F_{n-2}$ is the recursive relation part

The sequence starts with 0 rather than 1.

Fibonacci Sequence List

The List of first 10 terms in the Fibonacci Sequence is

$0,1,1,2,3,5,8,13,21,34$

The list of Fibonacci numbers are calculated as shown below.

$F_{n}$	$F_{n}=F_{n-1}+F_{n-2}$	$Fibonacci\ Number$
$F_{0}$	-	$0$
$F_{1}$	-	$1$
$F_{2}$	$F_{2}=F_{1}+F_{0}$	$1$
$F_{3}$	$F_{3}=F_{2}+F_{1}$	$2$
$F_{4}$	$F_{4}=F_{3}+F_{2}$	$3$
$F_{5}$	$F_{5}=F_{4}+F_{3}$	$5$
$F_{6}$	$F_{6}=F_{5}+F_{4}$	$8$
$F_{7}$	$F_{7}=F_{6}+F_{5}$	$13$
$F_{8}$	$F_{8}=F_{7}+F_{6}$	$21$

Applications of Fibonacci Sequence

• Used in financial analysis to identify trends in stock prices and other financial data
• Used to model various phenomena in biology, such as the growth patterns of plants and the arrangement of leaves on a stem.
• Used in Coding (computer algorithms, interconnecting parallel, and distributed systems)
• Used in numerous fields of science including high-energy physical science, quantum mechanics, Cryptography, etc.