1. The Fibonacci sequence

Some number patterns are invented by choosing a rule. Others seem to turn up everywhere once you know what to look for.

This course begins with one of the most famous: the Fibonacci sequence.

Build the sequence

Start with 1 and 1. Every new number is the sum of the previous two.

1, 1, 2, 3, 5, 8, 13, 21, 34, …

Definition
The Fibonacci rule
If we call the numbers F1, F2, F3, … then F1 = 1, F2 = 1 and each later number satisfies Fn = Fn−1 + Fn−2.
💡
Learning tip The little numbers underneath the F are just labels.

F1 means “the first Fibonacci number”, F2 means “the second Fibonacci number”, and so on.

So the rule simply means: each new number is made by adding the previous two.

Watch the sequence grow

Open each step to see how the next Fibonacci number is created.

1 + 1

1 + 1 = 2

Sequence: 1, 1, 2

1 + 2

1 + 2 = 3

Sequence: 1, 1, 2, 3

2 + 3

2 + 3 = 5

Sequence: 1, 1, 2, 3, 5

3 + 5

3 + 5 = 8

Sequence: 1, 1, 2, 3, 5, 8

5 + 8

5 + 8 = 13

Sequence: 1, 1, 2, 3, 5, 8, 13

8 + 13

8 + 13 = 21

Sequence: 1, 1, 2, 3, 5, 8, 13, 21

13 + 21

13 + 21 = 34

Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34

Practice puzzle
What comes next?

The sequence has reached 34.

1, 1, 2, 3, 5, 8, 13, 21, 34, …

Work out the next five Fibonacci numbers.
💡
Learning tipStart with the last two numbers you know: 21 and 34. Add them to get the next number, then repeat.
Reveal answer

The next five numbers are:

55, 89, 144, 233, 377

For example, 21 + 34 = 55, then 34 + 55 = 89.

A staircase puzzle

Suppose you climb a staircase by taking either one step or two steps at a time. How many different ways can you reach the top?

Explore
How many ways?

Try staircases with 1, 2, 3, 4 and 5 steps. Count carefully.

What sequence do your answers make?
💡
Learning tipFor the last move, you must have come from either one step below or two steps below.
Reveal answer

The numbers of routes are 1, 2, 3, 5, 8, … — Fibonacci numbers again.

Why? Every route to step n ends with either a 1-step move from step n−1 or a 2-step move from step n−2. So the number of routes obeys the same addition rule.

What have we learned?

  • The Fibonacci sequence starts 1, 1 and each term is the sum of the previous two.
  • The same recurrence can appear in counting problems.
  • A simple local rule can produce a surprisingly rich global pattern.

Next we will turn these numbers into shapes.

MAGICAL NUMBERS COURSE
Keep exploring
← Previous lesson
Back to Magical Numbers
Next lesson →