The Fibonacci Series



The Fibonacci grouping is a number arrangement characterized by a primary direct repeat connection. In arithmetic and science, the succession shows up in numerous settings.

Each being living in this world offers a firm example. A few mathematicians accept that we can locate a specific example in anything on the off chance that we look profoundly enough.

What does the winding game plan of leaves on a pine cone share practically speaking with the course of action of florets in a focal point of a daisy or the spirals of natural product buds on a pineapple? Those spirals can be approximated by a numerical arrangement of numbers known as Fibonacci numbers.

Here we will see those numbers that are not known by and large but exist in nature and utilized broadly. This article acquaints you with the Fibonacci Number arrangement and the basic meaning of the entire endless arrangement.


Fibonacci was born in the year 1170 to Guglielmo to an Italian vendor and customs official. Fibonacci went with his dad as a young boy, and it was in Bugia where he was instructed that he found out about the Hindu–Arabic numeral framework.

Before long, he understood the nHindu-Arabic framework’s umerous focal points, which, not at all like the Roman numerals utilized at that point, permitted simple figuring utilizing a spot esteem framework.

In 1202, he finished the Liber Abaci (Book of Abacus Liber Abaci, distributed in 1202 after Fibonacci Italy. The book depended on the number-crunching and variable based math that Fibonacci had gathered during his movements.

Fibonacci was born in the year 1170The book, which proceeded to be generally duplicated and imitated, presented the Hindu-Arabic spot esteem decimal framework and the utilization of Arabic numerals into Europe. Surely, albeit principally, a book about the utilization of Arab numerals, which is known as algorism, concurrent straight conditions are also concentrated in this work.

Fibonacci was a visitor of Emperor Frederick II, who took a keen interest in arithmetic and science. In 1240, the Republic of Pisa credited Fibonacci with giving him compensation in a declaration that remembered him for the administrations he had given to the city as a consultant on bookkeeping issues and guidance to residents.

Fibonacci died between the year 1240 and 1250 in Pisa.

The Fibonacci Series

The Fibonacci Sequence is the arrangement of numbers:

\[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...\]

The next number is calculated by adding the two numbers before it:

the 2 is found by adding the two numbers before it (1+1),

Similarly, the 3 is found by adding the two numbers before it (1+2),

the 5 is (2+3),


Fibonacci SeriesMany counting problems cannot be solved so easily. Suppose the number of bacteria in a colony doubles every hour. If a colony begins with five bacteria, how many will be present in n hours ?

To solve this problem,

Let an be the number of bacteria at the end of n hours. Since the number of bacteria doubles every hour, the relationship an = 2 an-1 holds whenever n is a positive integer. This relationship, together with the initial condition a0=5, uniquely determines an for all non negative integer n.

The counting problems can be solved by finding relationships, called recurrence relations between the terms of a sequence. The rule of finding terms from those that precede them is called a Recurrence relation.

A recurrence relation defines the sequence Fn of Fibonacci numbers

\[F_n = F_{n-1} + F_{n-2}\]

where n ∈ N (Natural number)

Fibonacci Ratio in Nature

The problem that Fibonacci researched in the year 1202 was about how quick bunnies could raise in ideal conditions.

The Rabbit's Problem

Expect an as of late imagined pair of rabbits, one male, one female, are set in a field. Bunnies can mate at one year of age so that at the completion of its second month a female can deliver another pair of rabbits. Expect that our rabbits never pass on and that the female reliably delivers one new pair (one male, one female) from the second month onwards. The enigma that Fibonacci introduced was...

What number of sets will there be in one year?

Let us analyze the problem

  • Close to the completion of the primary month, they mate. Anyway, there is so far only 1 set.
  • Close to the completion of the second month, the female makes another pair, so now there are 2 arrangements of rabbits in the field.
  • Around the completion of the third month, the principal female conveys an ensuing pair, making 3 sets in all in the field.
  • Around the fourth month’s completion, the primary female has made all the more new pairs; the female brought into the world two months earlier, delivering her first pair, making 5 sets.

Would you be able to perceive how the arrangement is shaped and how it proceeds?

Fibonacci Ratio in Nature
Source: Everything under the sky, Youtube

The number of pairs of rabbits in the field at the start of each month is \(1, 1, 2, 3, 5, 8, 13, 21, 34, …\)

Now we can understand why this series is the answer to our Rabbits problem.

Fibonacci numbers and the Golden Ratio

On the off chance that we take the ratio of two progressive numbers in Fibonacci's arrangement, (1, 1, 2, 3, 5, 8, 13, ..) and we divide each by the number before it, we will locate the accompanying arrangement of numbers:

\(\frac{1}{1} = 1, \frac{2}{1} = 2, \frac{3}{2} = 1·5, \frac{5}{3} = 1·666..., \frac{8}{5} = 1·6, \frac{13}{8} = 1·625, \frac{21}{13} = 1·61538...\)

It is simpler to perceive what's going on the off chance that we plot the proportions on a graph.

Fibonacci numbers and the Golden Ratio

The proportion is by all accounts settling down to a specific value, which we call the Golden ratio or the Golden number. It has an estimation of around 1·618034. The Golden ratio 1.618034 is likewise called the golden area or the golden mean or simply the Golden number. It is frequently represented by a Greek letter PhiThe firmly related value which we compose as phi with a little "p" is only the decimal piece of Phi, in particular 0·618034.

The Fibonacci sequence in nature

Observing the geometry of plants, flowers or fruit, it is easy to recognize the presence of recurrent structures and forms.

Fibonacci Numbers, the Golden Section and Plants

Watching the geometry of plants, blossoms or fruits, it is anything but difficult to perceive the presence of repetitive structures and forms.

One plant specifically shows the Fibonacci numbers in the quantity of "developing focuses" that it has. Assume that when a plant puts out another shoot, that shoot needs to grow two months before it is sufficiently able to help stretching. On the off chance that it branches each month after that at the developing point, we get the image appeared here.

Fibonacci Numbers, the Golden Section and Plants
Source :

A plant that develops especially like this is the "sneezewort": Achillea ptarmica.

The Fibonacci grouping, for instance, assumes an imperative part in phyllotaxis, which considers the course of action of leaves, branches, blossoms or seeds in plants, with the principle point of featuring the presence of ordinary examples. The different courses of action of characteristic components follow astonishing numerical normalities: D'arcy Thompson saw that the plant realm has an inquisitive inclination for specific numbers and for certain winding calculations, and that these numbers and calculations are firmly related.

We can undoubtedly discover the quantities of the Fibonacci grouping in the spirals framed by singular blossoms in the composite inflorescences of daisies, sunflowers, cauliflowers and broccoli.

the sunflower, individual flowers are arranged along curved lines

In the sunflower, individual flowers are arranged along curved lines which rotate clockwise and counterclockwise.
Source :

In the sunflower, individual flowers are arranged along curved lines which rotate clockwise and counterclockwise.

Another straightforward model in which it is conceivable to discover the Fibonacci grouping in nature is given by the number of petals of flowers. Most have three (like lilies and irises), five (parnassia, rose hips) or eight (cosmea), 13 (a few daisies), 21 (chicory), 34, 55 or 89 (asteraceae). These numbers are important for the celebrated Fibonacci sequence.

Lilies roses, petals, daisies

Applications of the Fibonacci Series

Fibonacci in Nature

Fibonacci spirals are additionally found in different fields related to nature. It is found in snails, shells, waves, blends of tones; roses and so forth in endless things made in nature.

Fibonacci in organs of human body

People show Fibonacci attributes. Each human has two hands, every last one of these has five fingers and each finger has three sections which are isolated by two knuckles. These numbers fit into the grouping. In addition the lengths of bones in a hand are in Fibonacci numbers.

Fibonacci in Music

The Fibonacci grouping of numbers and the brilliant proportion are shown in music generally. The numbers are available in the octave, the fundamental unit of melody and harmony

Fibonacci in Coding

In case you're a developer, there is a decent possibility you have executed a capacity to register the nth number of the Fibonacci succession eventually. It is frequently used to acquaint new developers with the idea of recursion and is a typical inquiry question.

There is additionally a calculation for looking through arranged exhibits that use Fibonacci numbers, called Fibonacci search. Moreover, Fibonacci is a well known decision for the scale utilized by Scrum groups for assessing work.

As of late Fibonacci arrangement and brilliant proportion are of extraordinary enthusiasm to the analysts in numerous fields of science including high energy physical science, quantum mechanics, Cryptography and Coding.

Fibonacci’s Other work

Fibonacci’s second work Practica geometriae was composed between 1220-1221. It relies on the works of the ancient Greek masters, including Euclid and Archimedes. Fibonacci also draws on the works of Plato of Tivloli (1145). The instructions were simple for measurement and effectively read tables are given where complex calculation would have been important to acquire the arrangement. Archimedes' technique for assurance of recorded and encompassed polygons is examined. Vague issues that follow from these lines of thought are likewise treated.

The third work Flos, made in 1225, was shipped off by Emperor Frederick II as a reaction to questions set out by Johannes of Palermo (a member from Emperor Frederick's court and a companion of Fibonacci's). Fibonacci gave solutions for second order and third order equations by providing a rational approximation of the solution. The third and final problem is the solution of a series of indeterminate equations.

In the Liber quadratorum created in 1225 Fibonacci acquired numerous striking accomplishments in number hypothesis. He gives an assortment of techniques for finding Pythagorean triples. He next experiences a definition for an uncommon class of numbers he calls a congruum, and afterward through splendid confirmation gets the full solution of the indeterminate equations.

These four works and a letter to Theodorus involve the enduring works of Fibonacci. These anyway are by all accounts not the only works formed by Fibonacci. Different works known to have existed incorporate the Di minor guisa, a book for business math. A conversation of Book X of Euclid's Elements is additionally known to have existed, in which Fibonacci gave a numerical conversation of irrational numbers.

Interesting Puzzles based on the series

Cumath Puzzles

Detailed Solution: The following table shows the maximum and final height of the snail each day.

table shows the maximum and final height of the snail each day

Continuing this way, we see that on Day-15, the maximum height of the snail will be 15 + 4 = 19 m. And the final height will be 19 - 4 =15m.

On Day-16, the snail will go up to 15 + 5 = 20m and, having got out of the pit, will not fall back. Which means the snail gets out of the pit on the 16th day.

Many problems are available in the Creative Reasoning section in Cuemath, including these types, and many many more.


The Fibonacci numbers are Nature's numbering framework. They show up wherever in Nature, from the leaf course of action in plants, to the example of the florets of a bloom, the bracts of a pinecone, or the sizes of a pineapple. The Fibonacci numbers are accordingly relevant to the development of each living thing, including a solitary cell, a grain of wheat, a hive of honey bees, and even the entirety of humankind. Nature follows the Fibonacci numbers incredibly. In any case, next to no we watch the excellence of nature. The Great writer Rabindranath Tagore additionally noticed this. In the event that we study the example of different characteristic things minutely we see that a considerable lot of the common things around us follow the Fibonacci numbers, in actuality, which makes bizarreness among us.

Frequently Asked Questions (FAQs)

Why is the Fibonacci sequence so important?

The centrality of the Fibonacci number arrangement is extraordinary. This sequence speaks to the diagrams of the all inclusive creation. This succession has verification that there was one big bang, however there is a period between them. It has a lot of numerical properties and many contributing applications to unsolved issues of number hypothesis. The Fibonacci arrangement reflects examples of development spirals found in nature. There is a fundamental calculation in the advancement of living things. Furthermore, that is significant.

What are the applications of the Fibonacci Sequence?

Fibonacci arrangements are surely found wherever in nature or life. They are valuable at demonstrating development of creature populaces, plant cell development, snowflake shape, plant shape, cryptography, and obviously software engineering. It is called the DNA pattern of nature.

How do you find the Fibonacci sequence?

In a Fibonacci sequence, you start with 0 and 1 and move on from there. Every latter entry in the sequence is produced by adding together the former two entries.

i.e. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. . . and so on









Is there a formula for the Fibonacci series?

The Fibonacci numbers are generated by setting F0 = 0, F1 = 1, and then using the recursive formula

Fn = Fn-1 + Fn-2

to get the rest. Thus the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

What is the golden ratio?

Golden ratio, also known as the golden section, golden mean, or divine proportion, in mathematics, the irrational number (1 + square root √5)/2, often denoted by the Greek letter ϕ or τ, which is approximately equal to 1.618.

If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ..) and we divide each by the number before it, The ratio seems to be settling down to a particular value, which we call the golden ratio or the golden number. It has a value of approximately 1•618034.

Where is the Fibonacci ratio in our body?

An intriguing relationship of Fibonacci proportion or golden section to the plan of the human body is that there are:

  • 5 extremities to the middle, in the arms, legs and head.
  • 5 extremities on each of these, in the fingers and toes
  • 5 openings on the face.
  • 5 receptors for sight, sound, touch, taste and smell.

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