Some number patterns are simple, but others hold a special kind of magic. The Fibonacci sequence is one of the most famous in all of mathematics. It starts with 0 and 1. From there, every new number is just the sum of the two that came before it.

Let's break it down:

• 0 + 1 = 1
• 1 + 1 = 2
• 1 + 2 = 3
• 2 + 3 = 5

And so on. The pattern continues forever, creating an endless string of numbers. This simple rule of adding the previous two numbers is the heart of the sequence. We can write this rule as a formula. If we call a term in the sequence $F_n$, where $n$ is its position, the rule is:

This is known as a recurrence relation, because each term is defined by the terms that came before it. To get it started, we just need to set the first two numbers, $F_0=0$ and $F_1=1$.

This sequence isn't a modern invention. It was introduced to the Western world in 1202 by an Italian mathematician named Leonardo of Pisa, who was known as Fibonacci. He included it in his book Liber Abaci, or "The Book of Calculation."

Fibonacci posed a thought experiment about the population of rabbits. Imagine you have a pair of newborn rabbits, one male and one female. Rabbits take one month to mature, and after that, a mature pair gives birth to another pair of rabbits every month. Assuming no rabbits ever die, how many pairs of rabbits will there be after one year?

At the start of month 1, you have one newborn pair. By month 2, they are mature, but haven't had babies yet, so you still have one pair. By month 3, they give birth to a new pair, so you have two pairs total. In month 4, the original pair has another set of babies, while the second pair matures. Now you have three pairs. The total number of pairs each month follows the sequence: 1, 1, 2, 3, 5, 8, ...

This simple puzzle gave the sequence its name and helped make it famous throughout history.

The Fibonacci sequence is more than just a party trick. It's filled with fascinating mathematical properties. One of the most remarkable is its connection to a special number called the golden ratio, often represented by the Greek letter phi ($\phi$). The value of $\phi$ is approximately 1.618.

If you take any number in the sequence and divide it by the one before it, the result gets closer and closer to $\phi$ as the numbers get larger.

• 8 ÷ 5 = 1.6
• 13 ÷ 8 = 1.625
• 21 ÷ 13 ≈ 1.615
• 144 ÷ 89 ≈ 1.618

Mathematically, this relationship is expressed as a limit:

Another interesting property is known as Cassini's Identity. It states that if you take three consecutive Fibonacci numbers, multiply the outer two, and then subtract the square of the middle one, the answer will always be either +1 or -1.

Let's test it with 5, 8, and 13. Here, $n$ is 6, since $F_6 = 8$.

$F_5 \times F_7 - F_6^2 = 5 \times 13 - 8^2 = 65 - 64 = 1$.

Since $n=6$ is even, $(-1)^6 = 1$. The identity holds true.

The sequence has deep connections to many areas of mathematics, from number theory to geometry. Its simplicity gives rise to surprising complexity, which is why mathematicians continue to study it. It serves as a foundational example of recursive patterns, which are crucial in computer science and other fields.