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Find the nth Term of a Linear Sequence

In this worksheet, students practise finding and using the nth term of a linear sequence.

'Find the nth Term of a Linear Sequence' worksheet

Key stage:  KS 4

Curriculum topic:  

Curriculum subtopic:  

Difficulty level:  

Worksheet Overview

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.  For example 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 will by now 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, it lets you find the value in the sequence if you know it's position.

For example, if you had an nth term of 4n - 5, you could find the 10th term by setting n=10 and substituting in to the nth term to get 4(10)-5 = 35.

The 10th term in the 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 numbers to be found.

 

Example 1: Find the nth term for 5, 7, 9, 11...

Step 1: Find the common difference. (a)

We can see that this sequence goes up 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. Just ask 'What number would come before the first term?'. In this example, the number would be 3 (the numbers change by 2 each time)

Our Nth term will be 2n + 3

 

Example 2: Find the nth term for -1, 3, 7, 11...

Step 1: Find the common difference. (a)

We can see that this sequence goes up 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 (the numbers change by 4 each time)

Our Nth term will be 4n - 5

 

Example 3: Find the nth term for 7, 4, 1, -2...

Step 1: Find the common difference. (a)

We can see that this sequence goes down 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 (the numbers change by 4 each time)

Our Nth term will be -3n + 10.

This could also be written as 10 - 3n

Complete the following sentence...

Complete the following sentence.

Match the sequence to its nth term.

Column A

Column B

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

Which one of these nth terms are correct for the sequence 5, 3, 1, -1...

 

-2n + 7

2n - 7

7 - 2n

-2n - 7

What is the nth term for 5, 10, 15...

 

Give your answer as simply as possible.

Find the nth term for the sequence 3, 8, 13, 18...

Find the nth term for -5, -3, -1, 1

Find the nth term for the sequence 6, 3, 0, -3....

Select if each nth term is correct or incorrect for the given sequence,

Which of these nth terms is the correct one for the sequence 5, 9, 13, 17...

4n + 1

-1 + 4n

4n - 1

1 - 4n

  • Question 1

Complete the following sentence...

CORRECT ANSWER
EDDIE SAYS
The nth term is a position to term rule. If you have the position, you can find the value of the number in that position.
  • Question 2

Complete the following sentence.

CORRECT ANSWER
EDDIE SAYS
Only linear sequences have this form. Quadratic, geometric etc will all have different forms
  • Question 3

Match the sequence to its nth term.

CORRECT ANSWER

Column A

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, all you need to do is look at the differences. 7, 10, 13, 16... for example has a difference of 3 so it must start with 3n (There's only one nth term that has this)
  • Question 4

Which one of these nth terms are correct for the sequence 5, 3, 1, -1...

 

CORRECT ANSWER
-2n + 7
7 - 2n
EDDIE SAYS
If you look at the differences, it is going down by 2. If there was another number at the front, it would be +7 This would give -2n + 7. Don't forget that this can also be written as 7 - 2n
  • Question 5

What is the nth term for 5, 10, 15...

 

Give your answer as simply as possible.

CORRECT ANSWER
5n
EDDIE SAYS
If you found the nth term as we have done before you would have 5n + 0. Is this in the simplest form possible?
  • Question 6

Find the nth term for the sequence 3, 8, 13, 18...

CORRECT ANSWER
5n-2
5n -2
5n- 2
5n - 2
EDDIE SAYS
This has a common difference of 5 and a zero term of -2. How do we put this together in an nth term?
  • Question 7

Find the nth term for -5, -3, -1, 1

CORRECT ANSWER
2n-7
2n -7
2n- 7
2n - 7
EDDIE SAYS
This has a common difference of 2 and a zero term of -7. How do we put this into the nth term?
  • Question 8

Find the nth term for the sequence 6, 3, 0, -3....

CORRECT ANSWER
-3n+9
-3n -9
-3n- 9
-3n - 9
9-3n
9 -3n
9- 3n
9 - 3n
EDDIE SAYS
There are actually two ways you can do this. The nth term you will probably have found first will be -3n + 9 but don\'t forget this can be written as 9 - 3n
  • Question 9

Select if each nth term is correct or incorrect for the given sequence,

CORRECT ANSWER
EDDIE SAYS
Remember that either the difference can be wrong or the zero term can be wrong. Q2 is going up in 4, but the zero term would be -1 Q3 is not going down in 5s
  • Question 10

Which of these nth terms is the correct one for the sequence 5, 9, 13, 17...

CORRECT ANSWER
4n + 1
EDDIE SAYS
This one has a common difference of +4 and a zero term of +1. How do we tie this together into a nth term.
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