**What is a linear sequence?**

A linear sequence is a sequence that has a **common difference**.

This means the sequence** increases or decreases by the same amount** each time.

**e.g. **4, 6, 8, 10, ... and 15, 11, 7, 3, ... **are** linear sequences but 1, 4, 9, 16, ... is **not**.

**What is an nth term?**

You may already be familiar with continuing a sequence by finding the difference - this is called a **term-to-term rule**.

An **nth term** is a **position-to-term** rule, which lets you find the value in the sequence if you know a term's position.

**e.g.** If we had an nth term for **4n - 5**, we could find the **10 ^{th}** term by setting

**n = 10**and substituting this in to the nth term to get:

**(4 × 10) - 5 = 35**

So the **10th term** in this sequence would be **35**.

**What do linear nth terms look like?**

All linear nth terms have the same structure **an + b**, where **a** and **b** are values to be found.

Let's look at these rules in action now in a real example.

**e.g. Find the nth term for this sequence: 5, 7, 9, 11, ...**

**Step 1**: Find the value of a (the common difference).

We can see that this sequence** increases by 2 **each time.

This means our nth term will start with **2n**.

**Step 2**: Find the value of b.

We can use a method here called 'the zero term'.

We need to ask ourselves: "What number would come before the first term?"

In this example, the number would be** 3**, as the terms are increasing by 2 each time.

So our nth term will be: **2n + 3**

Let's try another to check we have this method locked down.

**e.g. Find the nth term for this sequence: -1, 3, 7, 11, ...**

**Step 1**: Find the value of a (the common difference).

We can see that this sequence** increases by 4** each time.

This means our nth term will start with **4n**.

**Step 2**: Find the value of b.

"What number would come before the first term?"

In this example, the number would be **-5**, as the terms are increasing by 4 each time.

So our nth term will be: **4n - 5**

Here's a final example which features a **decreasing** sequence.

**e.g. Find the nth term for this sequence: 7, 4, 1, -2, ...**

**Step 1**: Find the value of a (the common difference).

We can see that this sequence** decreases by 3 **each time.

This means our nth term will start with **-3n**.

**Step 2**: Find the value of b.

"What number would come before the first term?"

In this example, the number would be **10**, as the terms are decreasing by 3 each time.

So our nth term will be: **-3n + 10**

This could also be written as: **10 - 3n**

In this activity, we will find the nth term for linear sequences in the format an + b, where a is the common difference and b is the zero term of the sequence.