Fibonacci number is a series of numbers where each number is the sum of two preceding numbers. In mathematics, the Fibonacci numbers are a series of numbers where a number is the addition of the previous two numbers, starting with \(0\) and \(1\). The recurrence relation gives it \({x_{n + 2}} = {x_n} + {x_{n + 1}}\). The Fibonacci sequence is given by: \(0,1,1,2,3,5,8,13,21,34,55,……….,\infty .\). This article aims to discuss all the details related to fibonacci numbers and will help students to understand the series better.

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Fibonacci Numbers: Introduction

A man puts a male-female pair of newly born rabbits in a field. Rabbits take a month to mature before mating. One month after mating, female gives birth to one male-female pair and then mate again. None of the rabbits die.

How many rabbit pairs are there after a year? For our convenience, we construct the table included below. At the start of each month, the number of young pairs, adult pairs, and total pairs is shown. For example, in the beginning of January, one pair of young rabbits is introduced into the population.

At the start of February, this pair of rabbits has matured. At the start of March, this pair has given birth to a new pair of young rabbits, and this process continues.

Month	January	February	March	April	May
Young	\(1\)	\(0\)	\(1\)	\(1\)	\(2\)
Adult	\(0\)	\(1\)	\(1\)	\(2\)	\(3\)
Total	\(1\)	\(1\)	\(2\)	\(3\)	\(5\)

Fibonacci Rabbit’s Population Table

We define the Fibonacci numbers \({F_n}\) to be the total number of rabbit pairs at the start of the \({n^{th}}\)  month. The number of rabbits pairs at the start of the \({13^{th}}\)month, \({F_{13}} = 233,\) can be taken as the solution to Fibonacci’s puzzle. Further examination of the Fibonacci numbers listed in the above table reveals that these numbers satisfy the recursion relation
\({F_{n + 1}} = {F_n} + {F_{n – 1}}\)

This recursion relation gives the following Fibonacci number as the sum of the previous two numbers. To start the recursion, we need to specify \({F_1}\) and \({F_2.}\) In Fibonacci’s rabbit problem, the initial month starts with only one rabbit pair so that \({F_1} = 1.\) And this initial rabbit pair is newborn and takes one month to mature before mating so \({F_2} = 1.\)

The first few Fibonacci numbers, read from the table, are given by \(1,1,2,3,5,8,13,21,34,55,89,144,233,….\) and has become one of the most famous sequences in mathematics.

Fibonacci Numbers: Definition

Fibonacci numbers are a series of numbers in which each Fibonacci number can be obtained by adding the two previous numbers. It means that the next number in the series is the sum of two previous numbers. For example, let the first two numbers in the series be taken as \(0\) and \(1.\) By adding \(0\) and \(1,\) we get the third number as \(1.\)

Then by adding the second and the third number, i.e., \(1\) and \(1,\) we get the fourth number as \(2,\) and similarly, the process goes on. Thus, we get the Fibonacci series as \(0,1,1,2,3,5,8,……. .\) Hence, the obtained series is called the Fibonacci number series.

We can also get the Fibonacci numbers from the pascal’s triangle as shown below:

Here, we can observe that the sum of diagonal elements represents the Fibonacci sequence, denoted by the lines.

Fibonacci Series: List

The list of numbers of a Fibonacci sequence is given below. This list is formed by using the recurrence relation
\({F_{n + 1}} = {F_n} + {F_{n – 1}}\)
Where \(n\) is the number of terms.

Fibonacci Number Series

\(0,\,1,\,1,\,2,\,3,\,5,\,8,\,13,\,21,\,34,\,55,\,89,\,144,\,233,\,377,\,610,\,987,\,1597,\) \(2584,\,4181,\,6765,\,10946,\,17711,\,28657,\,46368,\,75025,\,121393\) \(196418,\,317811,….\)

Fibonacci Numbers: Formula

The formula to obtain the Fibonacci numbers can be defined as:
\({F_{n + 1}} = {F_n} + {F_{n – 1}}\)
Where \({F_n}\) is the \({n^{th}}\) term or number
\({F_{n – 1}}\) is the \({\left({n – 1} \right)^{th}}\) term
\({F_{n – 2}}\) is the \({\left({n – 2} \right)^{th}}\) term

From the above formula, we can summarize the definition as the next number in the sequence is the sum of the previous two numbers present in the sequence, starting from \(0\) and \(1.\) So, let us create a table to find the next term of the Fibonacci sequence, using the above Fibonacci formula.

\({F_{n – 1}}\)	\({F_{n – 2}}\)	\({F_n}\)
\(0\)	\(1\)	\(1\)
\(1\)	\(1\)	\(2\)
\(1\)	\(2\)	\(3\)
\(2\)	\(3\)	\(5\)
\(3\)	\(5\)	\(8\)
\(5\)	\(8\)	\(13\)
\(8\)	\(13\)	\(21\)
\(13\)	\(21\)	\(34\)
\(21\)	\(34\)	\(55\)
\(34\)	\(55\)	\(89\)

In the above table, we can see the numbers in each column are relational and diagonal numbers are the same in all three columns.

The Fibonacci Spiral and The Golden Ratio

In Mathematics, two quantities are said to be in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The golden ratio is approximately \(1.618034.\)

The Fibonacci sequence is often visualized in a graph such as the Fibonacci spiral. Each of the squares explains the area of the next number in the sequence.  The Fibonacci spiral can be drawn inside the squares by connecting the corners of the boxes.

The squares fit together perfectly because the ratio between the numbers in the Fibonacci sequence is very close to the golden ratio, which is approximately \(1.618034.\) The larger the numbers in the Fibonacci sequence, the closer the ratio is to the golden ratio. The spiral and resulting rectangle are also known as the Golden Rectangle.

The spiral and resulting rectangle are also known as the Golden Rectangle.

Practice Exam Questions

Fibonacci Numbers: Calculation

The Fibonacci Sequence is closely related to the value of the Golden Ratio. We know that the Golden Ratio value is approximately equal to \(1.618034.\) and is denoted by the symbol \(\varphi .\) If we take the ratio of two successive Fibonacci numbers, the ratio is closely related to the Golden ratio.

For example, \(5\) and \(8\) are the two successive Fibonacci numbers. The ratio of \(8\) and \(5\) are:
\(\frac{8}{5} = 1.6\)

Take another pair of numbers, say \(13\) and \(21,\) the ratio of \(21\) and \(13\) is:
\(\frac{{21}}{{13}} = 1.6153\)
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 the Golden Ratio, we can find the Fibonacci numbers in the sequence.
The formula to calculate the Fibonacci numbers using the Golden Ratio is:
\({x_n} = \frac{{\left[{{\varphi ^n} – {{\left({1 – \varphi } \right)}^n}} \right]}}{{\sqrt 5 }}\)
Where, \(\varphi \) is the Golden Ratio, which is approximately equal to the value \(1.618\) \(n\) is the \({n^{th}}\) term of the Fibonacci sequence.

Fibonacci Numbers: Properties

The properties of the Fibonacci numbers are given below:

1. In the Fibonacci series, if we take any three consecutive numbers and add those numbers. When we divide the result by \(2,\) we will get the third number.
   For example, take three consecutive numbers such as \(2,3\) and \(5\) when adding these numbers, i.e., \(2 + 3 + 5 = 10.\) When \(10,\) is divided by \(2,\) the result is \(5,\) which is the third consecutive number \(5\)

2. Take four consecutive numbers other than \(0\) in the Fibonacci series. Multiply the outer number and also multiply the inner number. When you subtract these numbers, you will get the difference \(1.\)
   For example, take four consecutive numbers such as \(2,3,5,8.\) Multiply the outer numbers, i.e. \(2 \times 8\) and multiply the inner number, i.e., \(3 \times 5.\) Now subtract these two numbers, i.e. \(16 – 15 = 1.\) Thus, the difference is \(1.\)

Solved Examples – Fibonacci Numbers

Q.1. Write the first \(6\) Fibonacci numbers starting from \(0\) and \(1.\)
Ans: As we know, the formula for a Fibonacci sequence is
\({F_{n + 1}} = {F_n} + {F_{n – 1}}\)
Where \({F_n}\) is the \({n^{th}}\) term or number
\({F_{n – 1}}\) is the \({\left({n – 1} \right)^{th}}\) term
\({F_{n – 2}}\) is the \({\left({n – 2} \right)^{th}}\) term

Since the first term and second term are known to us, i.e. \(0\) and \(1.\) Thus,
\({F_0} = 0\) and \({F_1} = 1\)
Hence, the third term, \({F_2} = {F_0} + {F_1} = 0 + 1 = 1\)
Fourth term, \({F_3} = {F_2} + {F_1} = 1 + 1 = 2\)
Fifth term, \({F_4} = {F_3} + {F_2} = 1 + 2 = 3\)
Sixth term, \({F_5} = {F_4} + {F_3} = 3 + 2 = 5\)

So, the first six terms of Fibonacci sequence are \(0,1,1,2,3,5.)

Q.2. Find the next term of the Fibonacci series: \(0,1,1,2,3,5,8,13,21,34.\)
Ans: Each next term of the Fibonacci series is the sum of the previous two terms.
Therefore, the required term is \(21 + 34 = 55\)
Hence, \(55\)is the next term of Fibonacci series: \(0,1,1,2,3,5,8,13,21,34.\)

Q.3. Find the Fibonacci number using Golden ratio when \(n = 6.\)
Answer: The formula to calculate the Fibonacci numbers using the Golden Ratio is:
\({x_n} = \frac{{\left[{{\varphi ^n} – {{\left({1 – \varphi } \right)}^n}} \right]}}{{\sqrt 5 }}\)
Where, \(\varphi \) is the Golden Ratio, which is approximately equal to the value \(1618\) \(n\) is the \({n^{th}}\) term of the Fibonacci sequence.
\(n = 6\)

Now, substitute the values in the formula, we get
\({x_n} = \frac{{\left[{{\varphi ^6} – {{\left({1 – \varphi } \right)}^n}} \right]}}{{\sqrt 5 }}\)
\({x_n} = \frac{{\left[{{{1.618}^6} – {{\left({1 – 1.618} \right)}^6}} \right]}}{{\sqrt 5 }}\)
\({x_6} = \frac{{\left[{17.942 – {{\left({0.618} \right)}^6}} \right]}}{{2.236}}\)
\({x_6} = \frac{{\left[{17.942 – 0.056} \right]}}{{2.236}}\)
\({x_6} = \frac{{17.886}}{{2.236}}\)
\({x_6} = 7.999\)
\({x_6} = 8\) (Rounded value)
The Fibonacci number in the sequence is \(8\) when \(n = 6.\)

Q.4. Find the Fibonacci number using Golden ratio when \(n = 3.\)
Ans: The formula to calculate the Fibonacci numbers using the Golden Ratio is:
\({x_n} = \frac{{\left[{{\varphi ^6} – {{\left({1 – \varphi } \right)}^n}} \right]}}{{\sqrt 5 }}\)
Where, \(\varphi \) is the Golden Ratio, which is approximately equal to the value \(1.618.\) \(n\) is the \({n^{th}}\) term of the Fibonacci sequence.
\(n = 3.\)

Now, substitute the values in the formula, we get
\({x_n} = \frac{{\left[{{\varphi ^6} – {{\left({1 – \varphi } \right)}^n}} \right]}}{{\sqrt 5 }}\)
\[{x_3} = \frac{{\left[{{{1.618}^3} – {{\left({1 – 1.618} \right)}^3}} \right]}}{{\sqrt 5 }}\)
\({x_3} = \frac{{\left[ {4.235 + {{\left( {0.618} \right)}^3}} \right]}}{{2.236}}\)
\({x_3} = \frac{{\left[ {4.235 + 0.236} \right]}}{{2.236}}\)
\({x_3} = \frac{{4.471}}{{2.236}}\)
\({x_3} = 1.999\)
\({x_3} = 2\) (Rounded value)

The Fibonacci number in the sequence is \(2\) when \(n = 3.\)

Q.5. Find the Fibonacci number when \(n = 4\) using the recursive relation.
Ans: As we know, the formula for a Fibonacci sequence is;
\({F_{n + 1}} = {F_n} + {F_{n – 1}}\)
Where \({F_n}\) is the \({n^{th}}\) term or number
\({F_{n – 1}}\) is the \({\left({n – 1} \right)^{th}}\) term
\({F_{n – 2}}\) is the \({\left({n – 2} \right)^{th}}\) term

Since the first term and second term are known to us, i.e., \(0\) and \(1.\) Thus,
\({F_0} = 0\) and \({F_1} = 1\)
Hence, the third term, \({F_2} = {F_0} + {F_1} = 0 + 1 = 1\)
Fourth term, \({F_3} = {F_2} + {F_1} = 1 + 1 = 2\)
Fifth term, \({F_4} = {F_3} + {F_2} = 1 + 2 = 3\)
So, \(3\) is the Fibonacci number when \(n = 4\) using the recursive relation

Summary

In this article, we have discussed the origin of the Fibonacci numbers by using the famous rabbit example, the definition of a Fibonacci number. We also saw the list of Fibonacci series, the formula to calculate the Fibonacci numbers, the Fibonacci spiral and the golden ratio, and solved examples.

Frequently Asked Questions (FAQ) – Fibonacci Numbers

Frequently asked questions related to fibonacci numbers is listed as follows:
Q.1. What are the first \(10\) Fibonacci numbers?
Ans: The first 10 Fibonacci numbers are given by: \(0,1,1,2,3,5,8,13,21,34\) 

Q.2. How do you calculate Fibonacci?         
Ans: We will calculate the Fibonacci by using the formula
\({F_ {n + 1}} = {F_n} + {F_{n – 1}}\)
Where \({F_n}\)is the \({n^{th}}\) term or number
\({F_{n – 1}}\) is the \({\left({n – 1} \right)^{th}}\) term
\({F_{n – 2}}\) is the \({\left({n – 2} \right)^{th}}\) term

Q.3. What are Fibonacci numbers used for?
Ans: Fibonacci numbers play a very important role in financial analysis. Here, the Fibonacci number sequence can be used to generate the ratios or percentages that are useful for business purpose.

Q.4. What are the numbers in the Fibonacci sequence?
Ans: The Fibonacci sequence contains the numbers starting from \(0\) and \(1\) as: \(0,1,1,2,3,5,8,13,21,34,55,….,\infty \)

Q.5. What is the Fibonacci of \(5\)?
Ans: The notation that we will use to represent the Fibonacci sequence is as follows:
\({F_1} = 1,\,{F_2} = 1,\,{F_3} = 2,\,{F_4} = 3,\,{F_5} = 5\)
Hence, the \({5^{th}}\) Fibonacci number is \(5.\)

We hope this detailed article on Fibonacci numbers helped you. If you have any doubts or queries, you can ask us in the comment section. Happy learning!