### ... like I'm 5 years old

Imagine a series of numbers where each number is the sum of the two preceding ones. That's the Fibonacci sequence. It starts with 0 and 1, and continues like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The pattern is pretty simple: just add the last two numbers to get the next one.

The sequence gets its name from the Italian mathematician Leonardo of Pisa, also known as Fibonacci. He introduced it to the Western world in his 1202 book "Liber Abaci," but the sequence had been previously described in Indian mathematics.

Think of it like this: you have two bags of apples. The first bag has 0 apples. The second bag has 1 apple. Now, for every new bag, you take all the apples from the previous two bags and put them in the new bag. That's the Fibonacci sequence.

### ... like I'm in College

The Fibonacci sequence is not just a random series of numbers. It has some fascinating properties and patterns. For instance, if you divide a number in the sequence by the number before it, you get a ratio that hovers around 1.618. This number is known as the Golden Ratio, a mathematical concept that appears in various areas of art, architecture, and nature.

In addition, the Fibonacci sequence is related to the Fibonacci Spiral, a series of quarter circles inside squares whose side lengths are Fibonacci numbers. This spiral shows up in various aspects of nature, from the arrangements of leaves on a stem to the shape of a Nautilus shell.

Suppose we have a box of Lego bricks. We pick one 1-stud brick and place it on the table - this represents the first number in the Fibonacci sequence, 1. Then we take another 1-stud brick and place it next to the first one - this represents the second number, also 1.

Now, to represent the next number in the sequence, we take a 2-stud brick (since 1+1=2) and place it next to the first two. For the next number, we combine a 2-stud and 1-stud brick to make a 3-stud brick. Then, to get a 5-stud brick, we combine the 3-stud and 2-stud bricks. We can continue this process to build bricks representing larger Fibonacci numbers.

This Lego representation helps visualize the additive nature of the Fibonacci sequence and how each number is built from the sum of the previous two.

### ... like I'm an expert

As a mathematician, you're aware that Fibonacci numbers have numerous applications and appear in many branches of mathematics. They're used in Euclidean geometry, number theory, and combinatorics, to name a few. For example, the number of ways to tile a board of size n with squares and rectangles is a Fibonacci number.

Fibonacci numbers also have a close relationship with the Lucas numbers, another integer sequence. They're also linked to the golden ratio, and this connection can be used to derive Binet's formula for finding the nth Fibonacci number.