Fibonacci

# Exploring the Fibonacci Sequence

We designed this activity to give learners a chance to explore the history, applications, and natural occurances of this special number pattern.

There are optional exercises and challenges included within the sections, alongside hints and answers.

Select a heading to get started.

Fibonacci was a medieval Italian mathematician whose wrote "Liber abaci" in 1202. This book of research introduced Europe to using the decimal number system we still use today.

This book also included looking at a variety of solutions to different mathematical problems. One of these saw the creation of a number sequence as a possible model of rabbits breeding. Even though this solution did not work, the sequence was later studied by various other mathematicians and eventually (more than 600 years later) named the Fibonacci sequence.

It is amazing that a mathematician who has had such an impact on our numbering system, is best known for an answer that turned out to be incorrect for the problem he was trying to solve. Unbeknown to him, this sequence correctly applies to a lot of other mathematical observations in nature.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34

These are the first ten numbers of the Fibonacci sequence.

Can you see the pattern?

The sequence starts with 0 and 1, then follows the rule that each number is the sum of the previous two.

0 + 1 = 1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21
13 + 21 = 34

Can you work out the next 3 numbers in the sequence?

• 21 + 34 = 55
• 34 + 55 = 89
• 89 + 55 = 144

So the next three numbers are: 55, 89, 144.

Any number that appears in this sequence is known as a Fibonacci number.

There are some mathematically interesting patterns that can be observed in this sequence.

What do you notice about the numbers we get when adding together any three consecutive (next to each) numbers in the sequence? Why?

Start with adding together the first three numbers (0, 1, 1), then the next three (2, 3, 5), repeat until you've done this for the first 12 numbers of the sequence.

What do all the answers have in common?

When adding together any three consecutive numbers in the Fibonacci sequence, you will always get an even number.

No matter how you select three neighbouring numbers from this sequence, there will always be two odd numbers and one even number, meaning that when we add them together you will always get an even number.

Can you find the pattern for when you select six consecutive Fibonacci numbers, add them all together and then divide by four?

Now, investigate what happens when you take any four consecutive numbers from the Fibanacci sequence, add the first and last together and divide the total by 2.

Let us use Fibonacci to solve a fun sheep problem.

A sheep speak only using the letters A and B.

There is a set rule to their sequence of words.

The sequence starts with the letter A.

The rules are that the next word changings every A in the previous to B, and every B to an A and B.

So, the first four words are: A, B, AB, BAB

Can you work out the next 6 words?

The sequence goes:

• A
• B
• AB
• BAB
• ABBAB
• BABABBAB
• ABBABBABABBAB
• BABABBABABBABBABABBAB
• ABBABBABABBABBABABBABABBABBABABBAB
• BABABBABABBABBABABBABABBABBABABBABBABABBABABBABBABABBAB

This sequence of words contains patterns in their lengths and the number of the letters A and B. You should be able to answer the following questions without working out the next word in the sequence.

How many letters will the next word have?

In the next word, how many of the letters will be A?

In the next word, how many of the letters will be B?