21 Jul 2024

# Fibonacci Sequence – Definition, History, Formula, List, And Its Uses

Have you ever counted the number of petals on the flower? If you start counting the number of intact petals, you will notice that the number of petals is a Fibonacci number, which is the number from the Fibonacci sequence. Most flowers have about 3, 5, 8, 13, 21 or more petals. The leaves of cacti and the seeds of sunflowers are arranged in both left and right-handed spirals, and the number of seeds and leaves in these spirals is also placed in the Fibonacci sequence.

We have two hands in both hands, and we have five fingers. Each finger is divided into three parts again, and these are Fibonacci numbers. While the lengths of bones in hands are also Fibonacci numbers, so what is exactly the Fibonacci sequence, and why is it seemingly everywhere

## Definition

The Fibonacci sequence is a series of numbers in which a given number results from adding the two numbers before it. So if you start it with ‘0’, the next number will be ‘1’, followed by again 1, followed by 2, followed by 3, (0, 1, 1, 2, 3, 5, 8, 13,21, 34, 55……..). If you focus, you can see that every number in this series or sequence obtained by adding the two preceding numbers.

So,           0    +    1    =  1

1    +     1    =  2

1    +     2    =  3………..

This simple number sequence or series is called the Fibonacci sequence, and the numbers present in the sequence are called the Fibonacci numbers. Mathematically, the Fibonacci number sequence represents the formula.

Where n>1, you can use this expression to find any ‘n’th digit in the sequence.

## Leonardo Of Pisa And Finonacci Sequence

This interesting sequence is widely related to the mathematician named Leonardo Pisano, also known as Fibonacci. He hails from the Republic of Pisa and is called Leonardo of Pisa. As he was considered one of the most talented mathematicians of the Middle Ages, during that time when Europeans were still using Roman numbers, while the Hindu Arabic mathematicians were using a different number system, which was more robust and efficient.

He was fascinated by the brilliance of the Hindu Arabic numeral system. Fibonacci brought this system to the Western world in 1202 through his current famous book, Liber Abaci. In the book, he showed and compared the Hindu Arabic numeral system with other systems like Roman numerals. He also described how the Hindu-Arabic system made the calculation faster and easier.

This book contains the earliest known description of the Fibonacci sequence outside India. It has been described in ancient Indian texts by mathematicians as early as the 6th century, so this sequence’s number of series is unique. It may seem just a series of numbers, but this sequence has been discovered and again rediscovered in various forms, not just in mathematics but in nature and everyday life.

## Fibonacci Sequence- The Golden Ratio

Suppose you have two quantities, A and B, in which A is more significant than B. Now, add the A and B with the denominator A. So if this ratio comes out to be equal to A and B, then you can say that A and B have a golden ratio, which appears by the Greek letter called phi (φ)

Fibonacci sequence and calculation of its ratio using the formula, you will notice all the Fibonacci numbers have the golden ratio value, which is close to 1.618033

 F(n) F(n-1) F(n)/F(n-1) 1 1 1 2 1 2 3 2 1.5 5 3 1.666666667 8 5 1.6 13 8 1.625 21 13 1.615384615 34 21 1.619047619 55 34 1.617647059

In geometry, when the golden ratio applies as the growth factor, you get a particular type of logarithmic spiral. In simple terms, a golden spiral gets wider and wider by the factor of phi for each quarter turn it takes. Some of the daily life examples are sea shells, golden spirals in nature, ocean waves, hurricanes, and snail shells are some of the natural examples. This golden ratio is used in art architecture; you can see in galaxies, human ears, etc.

## Conclusion

The Fibonacci sequence has many interesting properties which have been studied for centuries. As we saw, each number is the sum of the previous two. The pattern starts from 0 and 1 and extends infinitely. Beyond the basic definition, the ratio of Fibonacci numbers comes nearer to the golden ratio as the numbers grow. This unexpected connection to geometry and patterns seen in nature is quite remarkable.

Fibonacci sequence can be seen in natural forms like spirals of shells, flower petals, seed heads, and tree branching points. Further applications of the Fibonacci sequence have also discovered in areas like stock market cycles and algorithmic trading strategies. Although defined, the Fibonacci sequence elegantly demonstrates how mathematical relationships exist in the biological world and other fields like economics. Its pervasive applications inspire continued research even today. In conclusion, the Fibonacci sequence is a quintessential example of beauty and utility through mathematics.