Fibonacci Sequence

Dhristi JEE 2022-24

What is the Fibonacci Sequence?

The Fibonacci sequence is a series of infinite numbers that follow a set pattern. The next number in the sequence is found by adding the two previous numbers in the sequence together. This can be expressed through the equation Fn = Fn-1 + Fn-2, where n represents a number in the sequence and F represents the Fibonacci number value


The Fibonacci sequence is seen everywhere in nature because it acts as a guide for growth. The Fibonacci Sequence is a series of numbers that starts with 0 and 1, and then each number in the sequence is equal to the sum of the two numbers before it.


Fibonacci Sequence = 0, 1, 1, 2, 3, 5, 8, 13, 21, ….


Here, “1” is the 3rd term and by adding the 1st and 2nd term we get 1. 


(i.e., 0+1 = 1)


Similarly,


By adding the 2nd and 3rd terms, we get 2 (1+1 = 2)


By adding the 3rd and 4th terms, we get 3 (1+2) and so on.


For example, the next term after 21 can be found by adding 21 and 13. 


Therefore, the next term will be 34.


What is the Fibonacci Series Formula?

The Fibonacci series numbers are in a sequence, where every number is the sum of the previous two. The first two are '0' and '1'.


To understand the Fibonacci series, we need to understand the Fibonacci series formula as well.


Fibonacci sequence of numbers is given by “Fn” 


It is defined with the seed values, using the recursive relation F₀ = 0 and F₁ =1:


Fn = Fn-1 + Fn-2


The sequence here is defined using 2 different parts, recursive relation and kick-off.


The kick-off part is F₀ = 0 and F₁ =1.


The recursive relation part is Fn = Fn-1 + Fn-2.


The sequence starts with the number '0'. So, F5 should be the sixth term in the sequence.


Golden Ratio to Calculate Fibonacci Numbers

The Fibonacci sequence can be approximated via the Golden Ratio. If consecutive Fibonacci numbers are of bigger value, then the ratio is very close to the Golden Ratio. In this way, we can find the Fibonacci numbers in the sequence. The Golden Ratio is approximately 1.618034. It's often denoted by the symbol φ. If you take the ratio of two successive Fibonacci numbers, it's close to the Golden Ratio. For example, the two successive Fibonacci numbers are 3 and 5. 


The ratio of 5 and 3 is:


5/3 = 1.6666


If we take another pair, say 21 and 34, the ratio of 34 and 21 is:


34/21 = 1.619


Formula to calculate Fibonacci numbers by Golden Ratio:


Xn = ɸn–(1−ɸ)n/√5


Where,


ɸ = Golden Ratio, which is approximately equal to the value 1.618


The nth term of the Fibonacci sequence is n.


Fibonacci Numbers Properties

Different algorithms use Fibonacci numbers (like Fibonacci cubes and the Fibonacci search technique), but we should remember that these numbers have different properties depending on their position.


The sequence series of Fibonacci can be extended to negative index n. The sequence is rearranged into this equation: 


Fn-2 = Fn - Fn−1


Fibonacci Sequence Calculator

The Fibonacci sequence is calculated within seconds by the free Fibonacci Calculators available online. The online calculator calculates are much faster than other methods and displays the sequence in a fraction of seconds. The procedure to use the tool is 


  • First in the input field enter the limit range. 

  • Next, click the “Find” button.  

  • The Fibonacci sequence will automatically be displayed in a new window.


Fibonacci Sequence Uses

The Fibonacci sequence is a series of numbers developed by Leonardo Fibonacci — a mathematician who was inspired by the patterns he found in nature and the everyday world. Whether we realize it or not, we can see patterns around us all the time: in math, art, and other areas of life. Understanding these patterns can help us predict behaviour and predict outcomes.


  • We can find Fibonacci numbers in the most common patterns and sequences of nature.

  • The Fibonacci sequence can be used to predict lunar eclipses, how leaf patterns appear on pineapple and even the formation of galaxies.

  • Sunflowers, seashells, and other organic or natural objects follow the same math that appears in the Fibonacci sequence.


The Fibonacci sequence facts reveal themselves in nature. One can observe them across natural and human creations.


Fibonacci Numbers Examples

1. When n=5, find the Fibonacci number, using recursive relation.


Solution - Fibonacci formula to calculate Fibonacci Sequence is


Fn = Fn-1+Fn-2


Take: F0=0 and F1=1


By using the formula, 


F2 = F1+F0 = 1+0 = 1


F3 = F2+F1 = 1+1 = 2


F4 = F3+F2 = 2+1 = 3


F5 = F4+F3 = 3+2 = 5


Therefore, the Fibonacci number is 5.

FAQs on Fibonacci Sequence

1. Why are Fibonacci series numbers important?

The Fibonacci numbers are most famously described as a sequence of integers where each number is the sum of the previous two numbers in the series. In most practical uses, including Calculus and other more complex mathematical subjects, this is how the numbers are applied as a ratio. But, they can be used more as a way to approximate and understand logarithmic spirals and how they work. One of the Fibonacci sequence's characteristics is that for any number in the sequence, the ratio of any number before it to the next tends toward a well-defined value.

2. Give an overview of the Fibonacci sequence?

The overview of the Fibonacci sequence is given below:


Leonardo Pisano Bogollo, an Italian, was the first to discover the Fibonacci sequence (Fibonacci). The Fibonacci sequence is made up of the numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The Fibonacci sequence is the name given to an endless series. Each word, starting at 0 and going up to 1, is the total of the two preceding ones. This is referred to as "nature's hidden code." Sunflowers, daisies, broccoli, cauliflowers, and seashells all have spiral designs that follow the Fibonacci sequence.

3. How does the puzzle of rabbits explain the Fibonacci sequence?

  • In the field, two newborn bunnies are left. At the conclusion of the first month, they are still one couple.

  • At the end of the second month, they mate and create a new pair, resulting in two pairs on the field.

  • The first couple gives birth to the second, but the second pair is left unbred, resulting in three pairs at the end of the third month.

  • The first pair generates a second pair, the second pair produces their first pair, and the third pair does not reproduce, resulting in a total of five pairs.

  • The cycle continues, and the number of rabbits in the field at the end of the nth month is equal to the sum of the number of mature pairs (n-2) and the number of pairs living last month (n-1). This is the number n in the Fibonacci sequence.

4. What are the applications of the Fibonacci sequence in the field of computer science?

The applications of the Fibonacci sequence in the field of computer science are:


  • The Fibonacci numbers play a crucial role in the computational run-time analysis of Euclid's technique for finding the greatest common divisor of two integers: the worst case input for this algorithm is a pair of successive Fibonacci numbers.

  • Some pseudorandom number generators employ Fibonacci numbers.

  • Fibonacci numbers appear in the Fibonacci heap data structure analysis.

  • Fibonacci numbers are used in a one-dimensional optimization method known as the Fibonacci search methodology.

  • In the IFF 8SVX audio file format for Amiga computers, the Fibonacci number sequence is employed for optional lossy compression. 

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