The Fibonacci Sequence Explained (and the Golden Ratio)

The Fibonacci sequence is one of the most famous number patterns in mathematics, and it comes from a rule a child could follow: each number is the sum of the two before it. From that tiny seed grows a sequence that connects to the golden ratio, appears throughout nature, and turns up constantly in number puzzles. This guide explains what the Fibonacci sequence is, the simple formula behind it, its surprising link to the golden ratio, and why it's worth recognising on sight when you're solving a number challenge.

What is the Fibonacci sequence?

The Fibonacci sequence is the list of numbers that starts with 0 and 1, where every following number is the sum of the previous two:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

So 0 + 1 = 1, then 1 + 1 = 2, then 1 + 2 = 3, then 2 + 3 = 5, and on it goes. The numbers themselves are called Fibonacci numbers. (Some versions start at 1, 1 instead of 0, 1, the pattern is identical from there.) The rule is so simple that you can extend the sequence as far as you like with nothing but addition.

The formula

Written compactly, the rule is:

F(n) = F(n−1) + F(n−2)

with the starting values F(0) = 0 and F(1) = 1. This is a recurrence: each term is defined using earlier terms, rather than a direct formula based on its position. (There is a direct formula, called Binet's formula, but it involves the golden ratio and square roots, the addition rule is what you'll actually use.)

Where it came from

The sequence is named after Leonardo of Pisa, known as Fibonacci, an Italian mathematician who introduced it to Western Europe in his 1202 book Liber Abaci. He posed it as a puzzle about breeding rabbits: starting with one pair, and assuming each pair produces a new pair every month, how many pairs are there after a year? The monthly counts follow the Fibonacci sequence. The numbers had been described earlier by Indian mathematicians, but Fibonacci's book made them famous in Europe.

The golden ratio connection

Here's the surprising part. If you divide each Fibonacci number by the one before it, the results get closer and closer to a single value:

5 ÷ 3 ≈ 1.667, 8 ÷ 5 = 1.6, 13 ÷ 8 ≈ 1.625, 21 ÷ 13 ≈ 1.615, ...

These ratios converge on the golden ratio, written φ (phi) and equal to about 1.618. The golden ratio is a special number that appears in geometry and art, and the deeper you go into the Fibonacci sequence, the closer consecutive ratios hug it. This link between a simple addition rule and a famous irrational number is one of the reasons the sequence has fascinated people for centuries.

Fibonacci numbers in nature

Fibonacci numbers really do appear in the natural world, though it's worth being precise rather than mystical about it. The number of petals on many flowers is often a Fibonacci number (3, 5, 8, 13). The spirals in a sunflower head, a pinecone, or a pineapple tend to come in counts that are consecutive Fibonacci numbers. The reason is mathematical: arranging seeds or leaves at an angle related to the golden ratio packs them efficiently without overlap. So the pattern isn't magic, it's the result of efficient growth, and not every flower follows it, but the tendency is real and widespread.

Fibonacci in number puzzles

In sequence puzzles, the Fibonacci pattern is a favourite trick. A sequence like 2, 3, 5, 8, 13 has no constant difference and no constant ratio, so it resists the usual checks, until you notice each term is the sum of the two before it. Recognising "sum of the previous two" on sight saves real time, which is why Fibonacci is one of the special sequences worth memorising alongside the squares and the primes. You'll meet it across the number challenges, especially at medium difficulty where recurrences first appear, and in our broader guide to the types of number patterns.

The Fibonacci sequence is proof that simple rules can produce deep, beautiful structure. Learn its opening terms, keep an eye out for the "sum of the last two" pattern, and you'll spot it everywhere, in puzzles and in the world.

Frequently asked questions

What is the Fibonacci sequence?

The Fibonacci sequence is a series of numbers where each term is the sum of the two before it, starting from 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on. The numbers are called Fibonacci numbers, and the rule is written F(n) = F(n−1) + F(n−2).

How is the Fibonacci sequence related to the golden ratio?

If you divide each Fibonacci number by the previous one, the ratios get closer and closer to the golden ratio, about 1.618. The further along the sequence you go, the more precisely consecutive Fibonacci numbers approximate this special value.

Where does the Fibonacci sequence appear in nature?

Fibonacci numbers commonly show up in the counts of flower petals and in the spirals of sunflowers, pinecones, and pineapples. This happens because arranging seeds or leaves at an angle linked to the golden ratio packs them efficiently. Not every plant follows it, but the tendency is genuine and widespread.

Who discovered the Fibonacci sequence?

It's named after Leonardo of Pisa, known as Fibonacci, who introduced it to Europe in his 1202 book Liber Abaci through a puzzle about breeding rabbits. The numbers had been described earlier by Indian mathematicians, but Fibonacci's work made them widely known in the West.