Fibonacci sequence

Some numbers follow a special rule.

To find the next number, you add the two before it. For example, one and one make two. Then, one and two make three.

A man named Fibonacci used this rule to talk about rabbits. He wanted to see how many rabbits there would be. These numbers appear in nature, too. You can see them in how leaves grow on a stem. They also show up in pine cones.

It is a pattern found in many places.

Imagine a pattern of numbers that grows using a simple rule. To find the next number, you add the two numbers before it.

Starting with 0 and 1, the list goes 0, 1, 1, 2, 3, 5, 8, and so on. These are called Fibonacci numbers.

This pattern is very old. People in India studied it as early as 200 BC. They used it to find patterns in poetry. Later, a man named Fibonacci shared it in Europe. He used the numbers to solve a puzzle about growing rabbits.

Nature loves these numbers too. You can see them in how leaves grow on a stem. They also appear in pine cones and artichokes.

These numbers are also linked to the golden ratio. This is a special number used in math. If you divide one Fibonacci number by the one before it, you get close to that ratio. This connection helps us understand many shapes in our world.

Imagine a list of numbers that grows by looking backward. To find a new number, you simply add the two numbers that came right before it. This special pattern is called the Fibonacci sequence. It usually starts with the numbers 0 and 1. If you add them, you get 1. If you add 1 and 1, you get 2. Then, 1 and 2 make 3, and 2 and 3 make 5. The list continues like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34.

This simple rule creates a sequence that can grow forever.

This pattern is much older than many people realize. Mathematicians in India studied these numbers as early as 200 BC. A person named Pingala used them to study patterns in Sanskrit poetry. He looked at how long and short sounds could be mixed together. Later, a writer named Virahanka showed how these patterns worked. In the year 1150, Hemachandra also wrote about these numbers.

These thinkers found the pattern in the rhythm of words long before it was famous in the West.

In 1202, an Italian mathematician named Leonardo of Pisa introduced the sequence to Europe. Most people know him by his nickname, Fibonacci. He used the numbers to solve a puzzle about rabbits. He imagined a field with a pair of rabbits that never die. He wanted to know how many pairs would be there after one year. The growing number of rabbit pairs followed the Fibonacci pattern perfectly.

His book, called The Book of Calculation, helped spread these ideas.

Nature often uses these numbers in surprising ways. You might see them in the way branches grow on a tree. They appear in the leaves on a stem and the fruit of a pineapple. You can even find them in the scales of an artichoke or a pine cone.

These numbers are also linked to something called the golden ratio. If you divide a Fibonacci number by the one before it, you get very close to this ratio. This connection helps explain many beautiful shapes in the world.

Today, these numbers are used in many important jobs. Computer scientists use them to create special tools called algorithms. They use them for things like the Fibonacci search technique. They also use them to build structures for computers to talk to each other.

There is even a special journal called the Fibonacci Quarterly just for studying them. From ancient poetry to modern computers, these numbers are everywhere.

The Fibonacci sequence is a famous list of numbers where each new number is the sum of the two numbers that came before it. This mathematical pattern is built using a simple recurrence relation. A recurrence relation is a rule that defines a sequence based on its previous terms. In this case, the rule is $F_n = F_{n-1} + F_{n-2}$. Most mathematicians start the sequence with 0 and 1. Following this rule, the sequence grows like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and 144.

This sequence is important because it appears in many different areas of science and math.

To understand how the sequence works, you can follow the steps of addition. You begin with two starting values, often called $F_0$ and $F_1$. To find the third number, you add the first and second together. To find the fourth number, you add the second and third together. This process continues indefinitely, creating a growing chain of numbers. You can even extend this sequence into negative numbers. By following the same rule in reverse, you can find numbers that come before zero. For example, if you follow the math backward, you can find values for negative indices. Most properties of these numbers stay the same whether the index is positive or negative.

While we call them Fibonacci numbers today, their history is very old. As early as 200 BC, mathematicians in India were using these numbers. A scholar named Pingala studied the patterns of Sanskrit poetry. He looked at how long and short syllables could be mixed to create rhythms. These patterns followed the Fibonacci sequence perfectly. Later, a writer named Virahanka provided a clear explanation of these patterns around 700 AD. By 1150, Hemachandra also wrote about how these numbers relate to poetic meters.

These early discoveries show that the pattern was understood long before it reached Europe.

In 1202, the Italian mathematician Leonardo of Pisa introduced the sequence to Western Europe. Most people know him by his nickname, Fibonacci. He shared these numbers in his book, *The Book of Calculation*. Fibonacci used the sequence to solve a specific puzzle about rabbit populations. He imagined an idealized group of rabbits that never died and always bred. He wanted to know how many pairs would exist after one year. The number of pairs at the end of each month followed the Fibonacci numbers exactly.

This example helped people see how math could model biological growth.

The Fibonacci sequence is deeply connected to a special number called the golden ratio. The golden ratio, often represented by the Greek letter phi, is a constant that appears in many shapes. There is a mathematical formula called Binet's formula that links these two ideas. This formula allows you to calculate any Fibonacci number using the golden ratio. Additionally, if you take two consecutive Fibonacci numbers and divide them, the result gets closer to the golden ratio as the numbers get larger.

This relationship is so important that Johannes Kepler observed it centuries ago. He noted that the ratios of these numbers almost perfectly match the golden ratio.

We can see these numbers working in the natural world. They appear in the way trees branch out and how leaves are arranged on a stem. You can find them in the patterns of a pineapple's fruit sprouts or the scales of an artichoke. Even the arrangement of bracts on a pine cone often follows these numbers.

These biological patterns are not just coincidences. They represent a way that nature organizes itself efficiently. While not every species uses this pattern, it is a common way for life to grow.

Today, Fibonacci numbers are used in advanced technology and computer science. Engineers use them to create the Fibonacci search technique, which is a way to find items in a list. They also use them to build a data structure called a Fibonacci heap. This structure helps computers manage information more effectively. There are even special graphs called Fibonacci cubes used to connect parallel and distributed computer systems. Because these numbers are so useful, there is even a scientific journal called the *Fibonacci Quarterly* dedicated to their study.