We always wonder why we have to learn and understand so many facts in math and where it gets used in real life, where we can see the implementation and examples related to the same. If you are curious enough you will find a lot of examples around you, for example look at the Fibonacci sequence the amount of similarity it consists with the world is just mind blowing. The answer behind nature’s secret code, golden ratio, and many more phenomena is Fibonacci sequence.

**Fibonacci**

A significant contribution to mathematics was made by Leonardo of Pisa, better known as Fibonacci. Through the centuries, his contributions to mathematics have intrigued and inspired people to explore the subject further. His name is best known for the sequence of numbers that bears his name, Fibonacci.

**Fibonacci Sequence**

A series of squares arranged in spiral form explains the appearance of the Fibonacci sequence, which is very beautiful and harmonious. The starting square has a side length of 1 and the square on its left has a side length of 1. Put another square on the two squares , and its side length is 2, then ,add the squares one by one, whose side length *[0,1,1,2,3,5,8,13,21,34,…..,n] *and so on. Every one of these numbers equals the sum of the former two numbers, and they just form the Fibonacci sequence. Many plants and animals show these numbers in their forms and designs, as well as in architecture, music, and art.

F_{n} | Fibonacci Number |

1 | 0 |

2 | 1 |

3 | 1 |

4 | 2 |

5 | 3 |

6 | 5 |

7 | 8 |

#### Formula

[0,1,1,2,3,5,8,13,21,34,…..,n]

From the sequence we can observe that

\( Fn=Fn-1 + Fn-2 \) for every \( n >1\).

\( F2=F1+F0\).

\( F3=F2+F1\).

\( F4=F3+F2\) and so on.

#### Example

Let’s find the Fibonacci number when n=4

Solution: By using the above formula \( Fn=Fn-1 + Fn-2 \)

Take: F_{0}=0 and F_{1}=1

we get

F_{2} = F_{1}+F_{0} = 1+0 = 1

F_{3} = F_{2}+F1 = 1+1 = 2

F_{4} = F_{3} + F_{2} = 2+1 = 3

The answer is 3 when number n=4 in Fibonacci sequence

#### Golden Ratio

Fibonacci Sequences are closely related to Golden Ratios. According to our knowledge, the Golden Ratio value is approximately 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. 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 is as follows \(large phi =frac{1+sqrt{5}}{2}\).

- F2/F1 = 1/1 = 1
- F3/F2 = 2/1 = 2
- F4/F3 = 3/2 = 1.5
- F5/F4 = 5/3 = 1.667
- F6/F5 = 8/5 = 1.6
- F7/F6 = 13/8 = 1.625
- F8/F7 = 21/13 = 1.615
- F9/F8 = 34/21 = 1.619
- F10/F9 = 55/34 = 1.617
- F11/F10 = 89/55 = 1.618

The Values of Golden Ratio is 1.618

Below is the diagrammatic representation of Fibonacci sequence

#### Fibonacci Sequence in Pascal’s triangle

Fibonacci numbers in Pascal’s Triangle The Fibonacci Numbers are also applied in Pascal’s Triangle. Entry is the sum of the two numbers either side of it, but in the row below. Diagonal sums in Pascal’s Triangle are the Fibonacci numbers.

In the below graph you can clearly see how the Fibonacci sequence is hidden in Pascal’s triangle.

Fibonacci sequence hidden in pascal’s triangle

#### Applications

Occurrence of Fibonacci Sequence can be seen and found almost everywhere within our universe. Below are some examples where it can be seen.

Petals of flowers is where it covers most of its’ property.

- 1 petal: white calla lily
- 3 petals: iris
- 5 petals: buttercup
- 8 petals: delphiniums
- 13 petals: cineraria

- used in the grouping of numbers and the brilliant proportion in music generally.
- used in Coding (computer algorithms, interconnecting parallel, and distributed systems)
- in numerous fields of science including high-energy physical science, quantum mechanics, Cryptography, etc.

#### Mind Blowing Properties

There are too many properties some of them are mentioned below:

- Summing together any ten consecutive Fibonacci numbers will always result in a number which is divisible by eleven
- Any two consecutive Fibonacci numbers are relatively prime, having no factors in common with each other
- Every third Fibonacci number is divisible by two,
- Every fourth Fibonacci number is divisible by three, or . Every fifth Fibonacci number is divisible by five and pattern continues
- If any two consecutive Fibonacci numbers are squared and then added together, the result is a Fibonacci number, which will form a sequence of alternate Fibonacci numbers.

#### Let’s Jump to Code

```
val Input = 15
println("The number is defined as: $Input")
fibonacciSeries(Input)
fun fibonacciSeries(Input: Int) {
var temp1 = 0
var temp2 = 1
println("The fibonacci series till $Input terms:")
for (i in 1..Input) {
print("$temp1 ")
val sum = temp1 + temp2
temp1 = temp2
temp2 = sum
}
}
```

#### Output

```
The number is defined as: 15
The Fibonacci series till 15 terms:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377
```

#### Conclusion

I think the Fibonacci sequence is the only pattern whose existence is too fascinating. Other than this I don’t think there is something of this magnitude present in our universe. It is a great example of how vast and interesting and fun math is which literally makes us believe that everything is possible when you have the vision of math.