Fibonacci Sequence

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.


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.

FnFibonacci Number



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.


Let’s find the Fibonacci number when n=4

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

Take: F0=0 and F1=1

we get

F2 = F1+F0 = 1+0 = 1

F3 = F2+F1 = 1+1 = 2

F4 = F3 + F2 = 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


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")
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


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


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.

Physics behind Free Fall


An object’s acceleration is the rate at which its velocity changes over time. The acceleration of an object is the result of all the forces acting on it, according to Newton’s second law. Gravity is the only force acting on a freely falling object under ideal circumstances. Calculate the displacement of an object that is free falling, calculate its average velocity at set intervals, and calculate its acceleration due to gravity.


When do we get to know if there is acceleration or not?

  • Drop something you pick up with your hand. As soon as you release it from your hand, its speed is zero. As it descends, its speed increases. As it falls, it travels faster. To me, that sounds like acceleration.
  • Acceleration is not just about increasing speed. Toss this same object vertically into the air. On the way up, its speed decreases until it stops and reverses direction. Acceleration also refers to a decrease in speed.
  • Acceleration involves more than just changing speed. One last time, launch your battered object. Throw it horizontally this time, and observe how its horizontal velocity gradually becomes vertical. Considering that acceleration is the rate at which velocity changes with time and velocity is a vector quantity, this change in direction is also considered acceleration.

Gravity was pulling the object down, so it was accelerating. The moment an object leaves your hand, it begins to fall – even if thrown straight up. In the absence of this, it would have continued moving away from you in a straight line. Gravity causes this acceleration.

Gravitational acceleration varies slightly with latitude and height above the earth’s surface. Sea level is experiencing greater acceleration due to gravity than Mount Everest at sea level and the poles are experiencing greater acceleration due to gravity at the equator. Free falling objects do not encounter a significant amount of air resistance; they fall solely under the influence of gravity. No matter how heavy an object is, it will fall with the same rate of acceleration. Galileo demonstrated with a demonstration that mass doesn’t really matter when it comes to free falling, contrary to what was previously thought.

Famous leaning tower of Pisa demonstration

This phenomenon was also observed by Galileo, who recognized that it was contrary to Aristotle’s principle that heavier items fall more quickly. According to an apocryphal tale, Galileo (or an assistant, more likely) dropped two objects of unequal mass from the Leaning Tower of Pisa. Contrary to Aristotle’s teachings, both objects struck the ground simultaneously (or very nearly so). Galileo probably couldn’t have obtained much information from this experiment given the speed at which the fall would occur. The majority of his observations of falling bodies were of round objects rolling down ramps. Water clocks and his own pulse were used to measure the time intervals (stopwatches and photogates hadn’t yet been invented). Until he reached an accuracy of one tenth of a pulse beat between two observations, he repeated this “a hundred times.”

The instant when the balls are released is considered to be the initial time t = 0. The position of the ball along the ruler is described by the variable y. The position of the ball at a time t is given by 

\(\large y(t)=y_0{}+v_0t+\frac{1}{2}g t^{2}\).

 If the ball is released from rest, the initial velocity is zero: v0 = 0. Therefore,

\(\large y(t)=y_0+\frac{1}{2}g t^{2}\).

Free fall bodies accelerate at -9.8 m/s (negative sign indicates downward acceleration). However, the acceleration of

A free falling body has a velocity of -9.8 m/s in the kinematic equation.

2) A drop from a particular height has an initial velocity of 0 m/s.

When an object is projected upwards vertically, it will slowly rise up. When it reaches 0 m/s, its velocity is zero

The apex of its trajectory.


 Calculate the body height if it has a mass of 1 kg and after 9 seconds it reaches the ground?


Given: Height h =?
Time t = 9s
We all are acquainted with the fact that free fall is independent of mass.

Hence, it is given as

\(\large h=\frac{1}{2}g t^{2}\).

h=0.5 * 9.8 * (9)2

h= 396.9

Falling with Air Resistance

It is common for an object to encounter some degree of air resistance as it falls through the air. Air molecules collide with the object’s leading surface, causing air resistance. There are a variety of factors that determine how much air resistance the object encounters. The speed of the object and its cross-sectional area are two factors that have a direct effect upon the amount of air resistance. An increase in speed results in an increase in air resistance. The amount of air resistance increases with an increase in cross-sectional area.

Common mistakes and misconceptions

A falling object’s final velocity is mistakenly assumed to be zero once it hits the ground. It is the speed just before touching the ground that determines the final velocity in physics problems. As soon as the object touches the ground, it ceases to be in freefall.

Math Coding

    fun height(gravity: Double, seconds: Double): Double {
        return 0.5 * gravity * Math.pow(seconds, 2.0)

    fun velocity(gravity: Double, seconds: Double): Double {
        return gravity * seconds

    fun time(velocity: Double, gravity: Double): Double {
        return velocity / gravity

    //Execution for Test Purpose
        val gravity = 9.80665
        var time = 10.0
        val height = height(gravity, time)
        val velocity = velocity(gravity, time)
        time = time(velocity, gravity)
        println("$velocity m/s")
        println("$time seconds")
        println("$height meter")



98.06649999999999 m/s
10.0 seconds
490.3325 meter


A free-falling object is studied using the kinematic equations. As a free falling object falls, it will be influenced by gravity. A free falling object has two main characteristics: it does not encounter air resistance, and it accelerates at a rate of 9.8 m/2. We can determine the height, velocity and time of free fall using the formula.