Find the Nth Term of a Linear Sequence

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.