There are a number of different types of **sequences**, some of which you may already be familiar with or have completed activities to learn more about.

The most common sequences are:

**Linear: **A sequence that **increases or decreases **by the same amount each time.** **

**Geometric**: A sequence that has a **common ratio**. This means we **multiply** by the same amount each time.

**Quadratic: **A quadratic sequence has a **common second difference**. This means that the **difference between the differences** will be the same.

Let's see these sequences in practice now.

**e.g. Describe the rule for, and give the next two terms of, the sequence 3, 6, 12, 24, ...**

We should notice that the sequence here is **doubling each time**.

This means the rule (or the common ratio) is **× 2**.

If we then continue the sequence using this rule, the next two terms will be:

5th: 24 × 2 =** 48 **

6th: 48 × 2 =** 96**

**e.g. Find the next term in the sequence 3, 6, 11, 18, 27, ...**

We first have to identify the sequence type here.

It isn't going up or down by the same amount so it can't be **linear**, but it also isn't multiplying each time, so it can't be a **geometric **sequence either.

This means it must be a **quadratic** sequence.

**Step 1: **Find the difference between the terms.

The difference between the first ans second terms is **3**.

The difference between the second and third terms is **5**.

The difference between the third and fourth terms is** 7**.

The difference between the fourth and fifth terms is** 9**.

Do you see how the difference is **going up by 2 **each time?

This means that the difference between the fifth and sixth terms is **11**.

If we add this to the fifth term, we will find our next term:

27 + 11 = **38**

In this activity, we will need to recognise all three types of sequences, find the rules which govern them, and apply these to find missing terms.

You should not use a calculator in this activity, but instead practise your mental arithmetic.