If you’re studying for your maths GCSE, you will have encountered a lot of different sequences. For both foundation and higher GCSE mathematics, you need to be able to identify these different types of sequences:**1. Linear sequences**

Linear sequences are the most common and simplest type of sequence you see in maths. You will have first come across these in primary school. They can simply be defined as sequences where the difference between each term is the same. Or another way of describing them is that the terms **add (or subtract) the same number each time.** Like the following sequences:

These are very easy to spot, and you also will need to be able to find the nth term of them for foundation and higher GCSE maths. For a guide on how to find the nth term of a linear sequence, read this blog.**2. Geometric progressions**Geometric sequences/progressions are similar to linear sequences, but instead of adding the same number each time, you

**multiply (or divide) by the same number each time**. Like the following sequences:

For both higher and foundation GCSE maths, you will need to be able to identify geometric progressions but do not need to find the nth term.**3. Fibonacci sequences**

This is probably the most famous sequence in mathematics. Fibonacci sequences are sequences where **each term is the sum of the previous two terms**. In a Fibonacci sequence, the first two numbers must be chosen before starting the sequence. The most common sequence begins with 0 and 1. This specific sequence of numbers generated by 0 and 1 are known as the “Fibonacci numbers”.

Since the first two terms are 0 and 1, the third term will be the first term and the second term added together (0 + 1) which gives 1. So the third term is also 1.

The fourth term will be the second and third terms added together (1 + 1) which gives 2.

The fifth term will be the third and fourth added together (1 + 2) which gives 3.

And so on…

However there are infinitely many Fibonacci sequences – that change depending on the two starting numbers chosen. So you can make up your own Fibonacci sequence – all you need to do is pick two starting numbers! Like the following sequence where I chose 2 and 5 as the starting numbers:

And this one where I chose 4 and 6:

There is no requirement to know how to find the nth term of a Fibonacci sequence in GCSE mathematics. However, in both foundation and higher Fibonacci sequences can come up. You should know how to identify them, understand how they work and find terms in the sequence.

To read more about the Fibonacci sequence and how it pops up in so many interesting and unexpected places, read our Fibonacci blog.**4. Quadratic sequences (higher tier only)**This is the most difficult type of sequence you will see in GCSE maths. Firstly, as you will be aware if you read the blogs on factorising quadratic expressions (foundation tier and higher tier), quadratics are expressions in which the highest power of x is an x

^{2}term. For example, 2x

^{2}+ 3x + 2 is a quadratic because the highest power of x is x

^{2}. However, x

^{3}– 2x

^{2}is not a quadratic, because although it contains an x

^{2}term, there is a higher power of x (the x

^{3}).

In quadratic sequences, the differences between the terms are not the same, however

**the difference of the differences are the same**. This is the only way you can identify them.

Take this example below. You see that the differences between the terms (in red) are different. However the differences of the red terms (in green) are the same.

And similarly in this example below. Again, the differences between the terms are different but the differences of the differences are the same.

It’s important to note that quadratic sequences only appear in the higher tier. In the higher tier, you will be expected to be able to identify quadratic sequences and find their nth terms. For a guide on how to find the nth term of a quadratic sequence, read this blog.

So those are the four types of sequence you need to be able to identify for GCSE maths. Now we know how to identify the four types, here are 20 sequences (10 each for foundation and higher). Put your answers in the comments or email them to sam@metatutor.co.uk and I’ll let you know if you are right. If you need some help with this, book in a free taster session.

Foundation tier (determine whether the sequences are linear, geometric or Fibonacci):

1. 5, 11, 17, 23, …

2. 3, 4, 7, 11, …

3. 6, 12, 24, 48, …

4. 1, 7, 8, 15, …

5. 17, 14, 11, 8, …

6. 32, 16, 8, 4, …

7. 21, 27, 33, 39, …

8. -4, -1, 2, 5, …

9. 2, 9, 11, 20, …

10. 1, 5, 25, 125, …

Higher tier (determine whether the sequences are linear, geometric, Fibonacci or quadratic):

1. 2, 6, 18, 54, …

2. 2, 7, 14, 23, …

3. 3, 8, 11, 19, …

4. 4, 5, 12, 25, …

5. 9, 13, 17, 21, …

6. 88, 44, 22, 11, …

7. 29, 21, 13, 5, …

8. 2, 5, 14, 29, …

9. -3, -6, -9, -12, …

10. 3, 10, 13, 23, …

what is the name of the sequence below or what is this sequence classified as?

1 4 9 13 17

Hi Crystal, that sequence doesn’t seem to fit in to any of the categories in this blog.

I would suggest that sequence is a typo, it looks like that 4 should be a 5 and it’s supposed to be a linear sequence going up in 4s – like this:

1 5 9 13 17

Sam