Fibonacci Series

This set of numbers follows a specific pattern, where each number is obtained by adding the two numbers before it. This sequence goes like 0, 1, 1, 2, 3, 5, 8, and so on.
 

The formula we use for the Fibonacci sequence is \(F(n) = F(n-1) + F(n-2)\) (where n is greater than 1). For example, the number 5 in the sequence is obtained by adding the terms 3 and 2 (applicable for every term).
 

Outside mathematics, the Fibonacci sequence appears in nature, design, and art. It can be observed in the branching patterns and the arrangement of their leaves.

History of the Fibonacci Sequence

The Fibonacci sequence is one of the revolutionary discoveries of an Italian mathematician, Leonardo Fibonacci. He wrote a book named Liber Abaci, which introduced numerous important concepts like the Fibonacci sequence, the Hindu-Arabic numeral system, and the decimal system.

Although it is said that this sequence originated years ago in Indian literature. Today, the Fibonacci sequence can be observed everywhere around us. Fibonacci patterns led to the development of a variety of designs and patterns. It has also been used in algorithms for searching and sorting tasks known as Fibonacci search.

The Fibonacci sequence formula helps us find any term in the Fibonacci series without listing all the numbers. It’s based on a simple rule, each term is the sum of the two terms before it.

If we start with
F₀ = 0 and F₁ = 1,
Then, each following term can be calculated using the recursive formula:

\(Fn = Fn-1 + Fn-2\), where n > 1.

This means:

• The 2nd term is found by adding the 1st and 0th terms.
• The 3rd term is found by adding the 2nd and 1st terms, and so on.
   

Let us look at an example: 
F₀ = 0, F₁ = 1
F₂ = 1 (0 + 1)
F₃ = 2 (1 + 1)
F₄ = 3 (1 + 2)
F₅ = 5 (2 + 3)

Hence, the Fibonacci sequence goes:
0, 1, 1, 2, 3, 5, 8, 13, 21, …

The Fibonacci numbers are unique and have special characteristics you might not know. Let’s explore these:

Recursive property: Each number in the sequence is the result of adding up the two preceding numbers. 
Example: 0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3, and so on. 


Golden ratio property: The ratio of any number to its preceding number approaches the golden ratio as the numbers get larger.  

   
Divisibility property:  
   

• For every 3rd Fibonacci number, it will be a multiple of 2. For example, 2, 8, 34, 144, etc. 
• For every 4th Fibonacci number, it will be a multiple of 3. For example, 3, 21, 144, etc. 
• For every 5th Fibonacci number, it will be a multiple of 5. For example, 5, 55, 610, etc. 
• For every 6th Fibonacci number, it will be a multiple of 8. For example, 8, 144, 610, 2584, 10946, etc. 
  
   

Sum of consecutive terms: The sum of any three consecutive Fibonacci numbers, when divided by 2, equals the third number. 
Example: \( 2+3+5=10\) and \(\frac{10}{2}=5\). 


Difference of the products: For any four consecutive numbers, the difference of the product of the outermost numbers and the inner numbers equals 1. 
Example: 1,2,3, and 5. 
1 × 5 = 5 (outermost numbers), 2 × 3 = 6 (innermost numbers). 
6 - 5 = 1.

The Fibonacci series spiral is a logarithmic spiral created by connecting the corners of squares whose side lengths follow the Fibonacci sequence. Each new square fits perfectly with the previous one, forming a smooth spiral that expands outward. This spiral pattern can be traced in many natural objects, such as sunflower seeds, snail shells, and the structures of hurricanes and spiral galaxies. The Fibonacci spiral captures how growth in nature often follows a balanced and proportional pattern.


Mathematically, this spiral is linked to the Golden Ratio (≈1.618), a unique number that represents perfect proportion and harmony. When a Fibonacci spiral is drawn inside a rectangle whose sides follow this ratio, it forms what is known as a golden rectangle, admired for its symmetry and natural beauty in both art and architecture.

Mathematically, this spiral is linked to the Golden Ratio (≈1.618), a unique number that represents perfect proportion and harmony. When a Fibonacci spiral is drawn inside a rectangle whose sides follow this ratio, it forms what is known as a golden rectangle, admired for its symmetry and natural beauty in both art and architecture.

In mathematics, the Fibonacci series and the Golden Ratio share a close and fascinating connection. The Fibonacci sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, … The relationship between them can be expressed by the formula:

\(Fn = \frac{(Φn - (1-Φ)n)}{√5}\),

where φ (phi) ≈ 1.618 represents the Golden Ratio.
 

The Golden Ratio can also be defined as the limit of the ratio between two consecutive Fibonacci numbers:
\(\phi = \lim_{n \to \infty} \frac{F_{n+1}}{F_n}\).

In simple terms, when you divide a Fibonacci number by the one before it, the result gets closer and closer to 1.618 as the numbers grow larger.

For example, 13 ÷ 8 = 1.625, 21 ÷ 13 = 1.615, 34 ÷ 21 = 1.619. As you can see, the ratio gradually approaches φ (1.618). It shows how the Fibonacci series naturally leads to the Golden Ratio.

An interesting way to find Fibonacci numbers is by using Pascal’s Triangle. In mathematics, Pascal’s Triangle is a triangular arrangement of binomial coefficients, where each number is the sum of the two numbers directly above it.

What’s fascinating is that Fibonacci numbers can be derived from this triangle by adding the numbers along its diagonals. If you start from the edge and move along the slanting diagonals, the sums of these diagonals form the Fibonacci sequence!

Fibonacci numbers vary in different types. These numbers follow a similar sequence, but the patterns may differ. Let’s learn the different ways to calculate the Fibonacci numbers. 

Recursive Relation Method: The sum of the two preceding numbers in the Fibonacci sequence. The formula for this is F(n) = F(n - 1) + F(n - 2). 
Finding 7th Fibonacci numbers
F(7) = F(6) + F(5)
= 8 + 5 = 13.
 

Golden Ratio Method: The Golden Ratio and the Fibonacci sequence are closely related. The symbol denotes it ɸ. The equation to find the Golden ratio is . 
 

Binet’s Formula (Closed-Form Expression):  To find the Fibonacci sequence using Binet’s formula, we use the formula F(n) = ɸn - (1 -ɸ )n / √5. Here, ɸ is the golden ratio, and n is the nth term of the Fibonacci sequence. 
 

Matrix Exponentiation: The Fibonacci sequence is the sum of the previous two Fibonacci numbers. Using a matrix makes it easy to calculate the sequence. The equation to find the nth Fibonacci number is 
 

We have now learned the applications of the Fibonacci sequence in various sectors. This set of numbers has tremendous importance in mathematics due to its special properties. The sequence frequently reveals a variety of mathematical patterns like the golden ratio and can be observed in geometry. Moreover, we can also use these numbers in problem-solving related to network structures.
 

Mastering the Fibonacci sequence is an important skill, but it can be a difficult task for students. We will now discuss a few tips and tricks to help you learn it easily:

• Students should recall that in the Fibonacci sequence, each number is the sum of the two numbers before it.

• Children can visualize the Fibonacci pattern in their daily lives to make it easier to understand. For example, think of the spirals in the seeds of sunflowers.

• They can practice learning the sequence using finger calculations and mental math, or can learn by using Fibonacci sequence calculator.

• Do not skip steps while solving problems related to the Fibonacci sequence.
  
   
• Teachers and parents can show children real-life examples of Fibonacci patterns, like the spirals in pine cones, shells, or sunflower heads. This helps them see the sequence as more than just numbers.
  
   
• Teachers and parents can help students create Fibonacci spirals using blocks, beads, or paper squares. Hands-on activities make the pattern easier to grasp.
  
   
• Let students identify Fibonacci patterns in art, architecture, or nature around them. Turn it into a discovery activity or a classroom project.
  
   
• Connect Fibonacci concepts with science (plant growth), art (design symmetry), and coding (patterns in algorithms).
  
   
• Parents and teachers can challenge students to predict the next number in the sequence or find Fibonacci numbers in Pascal’s Triangle.

The Fibonacci sequence has paramount importance in different sectors. Understanding its real-world applications can help them understand the different number patterns around them.

• In Nature and biology: The Fibonacci sequence is widely present in various forms in nature like petal arrangements, plant growth and shapes. It can be observed in the specific number of petals on many flowers (e.g., 3, 5, 8, 13, etc.). The patterns appear in the branching of trees and the arrangement of leaves on a stem are in Fibonacci sequence. Also, this sequence is key to the spiral arrangement of seeds in a sunflower and the spirals seen in pine cones, hurricanes, and shells. 
  
   
• In art, architecture, and design: The Fibonacci sequence and its related concept, the Golden Ratio ($\phi$), have been a source of inspiration for numerous patterns in art and design. The Golden Ratio is a crucial concept used to design important architectural structures due to its visually pleasing proportions. The sequence and the Golden Ratio can be observed in famous artworks. For example, Da Vinci's Vitruvian Man showcases Golden Ratio proportions.
  
   
• In finance and technology: The sequence also has technical applications in modern sectors. The sequence is utilized in finance to analyze markets like the stock market. Traders use Fibonacci numbers to determine possible rates of support and resistance. Fibonacci numbers are used in computer programs to improve efficiency in algorithms for sorting and searching tasks.