# Find the Nth Term of a Linear Sequence

In this worksheet, students will find and utilise 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.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Algebra

Curriculum subtopic:   Sequences

Difficulty level:

### QUESTION 1 of 10

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 10th 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.

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.

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.

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.

Complete the sentence below to define what an nth term is.

Complete the sentence below to summarise a particular type of sequence.

Match each sequence below to its correct nth term.

## Column B

2, 3, 4, 5, ...
n + 1
7, 10, 13, 16, ...
-6n + 23
4, 6, 8, 10, ...
3n + 4
17, 11, 5, -1, ...
2n + 2

Which of the nth terms below correctly matches this sequence?

5, 3, 1, -1, ...

-2n + 7

2n - 7

7 - 2n

-2n - 7

What is the nth term for this sequence?

5, 10, 15, ...

What is the nth term for this sequence?

3, 8, 13, 18, ...

Give your answer as simply as possible and without any spaces to ensure that you are marked correctly.

What is the nth term for this sequence?

-5, -3, -1, 1, ...

Give your answer as simply as possible and without any spaces to ensure that you are marked correctly.

What is the nth term for this sequence?

6, 3, 0, -3, ...

Give your answer as simply as possible and without any spaces to ensure that you are marked correctly.

Select if each nth term given below is correct or incorrect for the sequence it appears with.

Which of the nth terms below correctly matches this sequence?

5, 9, 13, 17, ...

4n + 1

-1 + 4n

4n - 1

1 - 4n

• Question 1

Complete the sentence below to define what an nth term is.

EDDIE SAYS
The nth term is a position-to-term rule. This means that if we have the position of the term we are seeking, we can find the value of the number in that position in a much quicker and easier way than continuously applying a term-to-term rule.
• Question 2

Complete the sentence below to summarise a particular type of sequence.

EDDIE SAYS
Only linear sequences use the form: an + b Other sequences (e.g. quadratic, geometric, etc.) use different formats to express their position-to-term rules.
• Question 3

Match each sequence below to its correct nth term.

## Column B

2, 3, 4, 5, ...
n + 1
7, 10, 13, 16, ...
3n + 4
4, 6, 8, 10, ...
2n + 2
17, 11, 5, -1, ...
-6n + 23
EDDIE SAYS
With this one, we need to calculate the differences in each sequence, then match these to the value of a in an + b. 2, 3, 4, 5, ... = increasing by 1 each time So this will match n + 1 as this is the only expression which has a value of 1 accompanying n. 7, 10, 13, 16, ... = increasing by 3 each time So this will match 3n + 4 as this is the only expression which has a value of 3 accompanying n. Can you follow this process independently to match the final two pairs of sequences to their nth terms?
• Question 4

Which of the nth terms below correctly matches this sequence?

5, 3, 1, -1, ...

-2n + 7
7 - 2n
EDDIE SAYS
If we look at the differences between the terms in this sequence, we can see it is decreasing by 2. If there was another number added before the first term (the zero term), this would be 7. If we put these two facts together, the simplest expression of the nth term for this sequence would be -2n + 7. Don't forget that this can also be written as 7 - 2n, as these two formats are identical. Did you spot that second option?
• Question 5

What is the nth term for this sequence?

5, 10, 15, ...

5n
EDDIE SAYS
The difference between the terms in this sequence is that they are increasing by 5 each time. If we add a term before our first term (the zero term), it would be 0. Therefore, if we put this together, we find that our nth term is 5n + 0. Is this in the simplest form possible? Adding 0 makes no difference to our overall total, so we can remove this to leave us with the simplest possible answer of 5n.
• Question 6

What is the nth term for this sequence?

3, 8, 13, 18, ...

Give your answer as simply as possible and without any spaces to ensure that you are marked correctly.

5n-2
5n -2
5n- 2
5n - 2
EDDIE SAYS
This sequence has a common difference of 5 and a zero term of -2. How do we put this together in an nth term? That's correct, we would express this as 5n - 2.
• Question 7

What is the nth term for this sequence?

-5, -3, -1, 1, ...

Give your answer as simply as possible and without any spaces to ensure that you are marked correctly.

2n-7
2n -7
2n- 7
2n - 7
EDDIE SAYS
This sequence has a common difference of 2 and a zero term of -7. How do we put these two elements together into an nth term?
• Question 8

What is the nth term for this sequence?

6, 3, 0, -3, ...

Give your answer as simply as possible and without any spaces to ensure that you are marked correctly.

-3n+9
-3n +9
-3n+ 9
-3n + 9
9-3n
9 -3n
9- 3n
9 - 3n
EDDIE SAYS
The common difference in this sequence is -3, and the zero term is 9. There are actually two ways we can express these elements as an nth term. If we put these together as usual, we reach -3n + 9. But don't forget this can also be written as 9 - 3n. Which way round do you prefer?
• Question 9

Select if each nth term given below is correct or incorrect for the sequence it appears with.

EDDIE SAYS
If an nth term does not match its sequence, then either the difference can be wrong or the zero term can be wrong. For the first sequence, the difference is - 2 and the zero term is 8 so this does match. For the second sequence, the difference is + 4 which does match, but the zero term should be -1 so this does not. For the third sequence, the difference is - 6 which does not match, but the zero term should be 6 so this does.
• Question 10

Which of the nth terms below correctly matches this sequence?

5, 9, 13, 17, ...

4n + 1
EDDIE SAYS
This sequence has a common difference of + 4 and a zero term of 1. How do we tie these elements together into a nth term? Congratulations on completing this activity! Hopefully you are feeling much more confident to find the nth term for a linear sequence in the form an + b.
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