What is the Fibonacci sequence?

Welcome! I am so excited to explore this beautiful pattern with you. Before we dive into the math and see how it builds everything from seashells to galaxies, I would love to learn a bit more about what brings you here.

To help me tailor our session, could you tell me a little about why you're interested in the Fibonacci sequence and how much you already know about it?

I'm a complete beginner and just curious about how it appears in nature!

It is wonderful to have you here! Since you are a beginner looking to spot these patterns in the wild, let's start with the simple rule that builds the entire sequence.

The Add-As-You-Go Rule

The is a string of numbers where you find the next number by simply adding up the two numbers that came right before it. We start our journey with just two numbers: 0 and 1.

F_n = F_{n-1} + F_{n-2}

Using this add-the-last-two rule, the sequence grows step by step:

0
1
1 (from 0 + 1)
2 (from 1 + 1)
3 (from 1 + 2)
5 (from 2 + 3)
8 (from 3 + 5)

This simple progression keeps going forever: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...

Why does this sequence start with 0 and 1?

That is such a natural question to ask! We use 0 and 1 as our starting blocks simply because they generate the most famous, standard version of the sequence, which we call the .

But here is the real secret: the magic isn't actually in the starting numbers themselves. The magic is in the rule of adding the last two numbers together. Because that rule is so powerful, you can actually start a sequence with any two numbers you want!

The Choose Your Own Adventure Sequence

To see how this works, let's build a brand-new sequence together using the exact same rule, but we will start with two completely different numbers: 2 and 4. Let's calculate the first few steps together:

We start with 2 and 4.
Next, we add them together: $2 + 4 = 6$ .
Now we add our last two numbers: $4 + 6 = 10$ .
Next step: $6 + 10 = 16$ .
And then: $10 + 16 = 26$ .

Even though we didn't start with 0 and 1, we still built a perfectly valid, beautiful mathematical sequence: 2, 4, 6, 10, 16, 26... Mathematicians call these family members "Fibonacci-like" sequences. In fact, if you start with 1 and 3 instead, you get a very famous sister sequence called the .

Starting with 0 and 1 is simply the standard convention. It gives us the cleanest baseline to study the pattern before we look at how these specific numbers shape the natural world.