# Arithmetic Sequences (the Nth Term)

In this worksheet, students find the Nth term in simple linear sequences.

Key stage:  KS 2

Curriculum topic:   Maths and Numerical Reasoning

Curriculum subtopic:   Sequences

Difficulty level:

### QUESTION 1 of 10

In an Arithmetic sequence like this:

4, 10, 16, 22, 28, 34 ...

the nth term is found as follows:

Step 1

Find the common difference between the terms

Here it is 6.

Step 2

The common difference is the coefficient (this just means the number in front.) of n

Here we get 6n

Step 3

Compare this to the actual sequence and adjust the formula by adding or subtracting a fixed number.

Here 6n would give us the terms 6, 12, 18, 24, 30 etc.... which is 2 more than our sequence of 4, 10, 16, 22, 28 etc.

So we must subtract 2.

So the nth term is 6n - 2

We can test this by putting in n = 5 to check that the 5th term is really 28.

It is, because 6 x 5 - 2 = 30 - 2 = 28

For this worksheet, It is important that you do not put spaces in your answer (For example, you would need to put in 5n-3 instead of 5n  -  3. If you put in spaces, the system will mark it as wrong.

Work out the common difference is in this sequence:

1, 2, 3, 4, 5, 6 ...

If the common difference in this sequence is two, what is the correct way of writing the nth term in this sequence?

2, 4, 6, 8, 10, 12 ...

n+2

n2

2n

2 + n

Work out the nth term in this sequence:

5, 10, 15, 20, 25, 30 ...

Remember to write your answer as a number followed directly by n.  For example, 2n

Work out the nth term in this sequence:

2, 3, 4, 5, 6, 7 ...

1, 3, 5, 7, 9, 11  ...

Select an answer in each group to give the correct answer for the nth term in this sequence.

Claire thinks that the nth term for this sequence is 3n -1.  Is she correct?

1, 4, 7, 10, 13, 16 ...

Correct

Incorrect

Work out the nth term in this sequence:

4, 7, 10, 13, 16, 19 ...

2n-1

3n+1

2n+2

3n-1

It's time for you to write the correct nth term for this sequence:

1, 6, 11, 16, 21, 26 ...

Remember, when writing don't include any spaces between your answers.  For example, 3n+1

Work out the nth term in this sequence:

10, 17, 24, 31, 38, 45 ...

Work out the nth term in this sequence:

9, 13, 17, 21, 25, 29 ...

• Question 1

Work out the common difference is in this sequence:

1, 2, 3, 4, 5, 6 ...

EDDIE SAYS
To find the common difference work out what number you need to add each time, to get to the next number in the sequence. Did you find the common difference was 1? 1 + 1 = 2 2 + 1 = 3 3 + 1 = 4 4 + 1 = 5
• Question 2

If the common difference in this sequence is two, what is the correct way of writing the nth term in this sequence?

2, 4, 6, 8, 10, 12 ...

2n
EDDIE SAYS
The common difference is two in this sequence. In maths, the way to write this down is to say that the nth term is 2n. Remember n represents the position of that number in the sequence. 2n really means (2 x position in the sequence). 2 (common difference) x 1 (position in the sequence)= 2 2 (common difference) x 2 (position in the sequence) = 4 2 (common difference) x 3 (position in the sequence) = 6 2 (common difference) x 4 (position in the sequence) = 8 2 (common difference) x 5 (position in the sequence) = 10 2 (common difference) x 6 (position in the sequence) = 12
• Question 3

Work out the nth term in this sequence:

5, 10, 15, 20, 25, 30 ...

Remember to write your answer as a number followed directly by n.  For example, 2n

5n
5 n
EDDIE SAYS
The common difference is 5 in this sequence. This means that we simply record this as 5n. If you put a space in between '5' and 'n', or you didn't write n at all, you would have been marked as incorrect. It's important to learn to write your answer in the correct mathematical way so that you get maximum marks in a test.
• Question 4

Work out the nth term in this sequence:

2, 3, 4, 5, 6, 7 ...

EDDIE SAYS
How did you do? If you are unsure why not have a look at the 3 steps in the introduction. First, we know that the sequence increases by 1 each time so the value for n must be 1. In maths, we would write this simply as n (as 1n simply means 1 x n which means exactly the same). But we need to work out what we need to do to n to get the next number in the sequence. 1 (position in the sequence) x 1 (value for n) = 1. We must + 1 to get to the number 2. 2 (position in the sequence) x 1 (value for n) = 2 We must + 1 to get to the number 3 3 (position in the sequence) x 1 (value for n) = 3 we must +1 to get to the number 4 4 (position in the sequence) x 1 (value for n) = 4 we must + 1 to get to the number 5 5 (postiion in the sequence) x 1 (value for n) = 5 we must + 1 to get to the number 6 So, our answer must be n+1. If you put 1n, we have allowed the mark but remember if n is 1, we just write that as n. You should give your answers as a number not a word.
• Question 5

1, 3, 5, 7, 9, 11  ...

Select an answer in each group to give the correct answer for the nth term in this sequence.

EDDIE SAYS
Are you getting the hang of this now? We know that the common difference is 2. So we can write this down as 2n. Now, we need to see if we should add or subtract anything to 2n to get the next number in the sequence. Let's break it down. 1 (position in the sequence) x 2 (value for n) = 2. We must - 1 to get to the number 1. 2 (position in the sequence) x 2 (value for n) = 4 We must - 1 to get to the number 3 3 (position in the sequence) x 2 (value for n) = 6 we must -1 to get to the number 5 4 (position in the sequence) x 2 (value for n) = 8 we must - 1 to get to the number 7 5 (postiion in the sequence) x 2 (value for n) = 10 we must - 1 to get to the number 9 Hopefully, this is starting to make more sense now?
• Question 6

Claire thinks that the nth term for this sequence is 3n -1.  Is she correct?

1, 4, 7, 10, 13, 16 ...

Incorrect
EDDIE SAYS
What did you think? Claire is incorrect with her answer. Somewhere, she has gone wrong in her process. Let's work out what the answer should be. We can see that the common difference between the numbers is 3. So, let's write down 3n and come back to this later. 1 (position in the sequence) x 3 (value for n) = 3. We must - 2 to get to the number 1. 2 (position in the sequence) x 3 (value for n) = 6 We must - 2 to get to the number 4 3 (position in the sequence) x 3 (value for n) = 9 we must -2 to get to the number 7 4 (position in the sequence) x 3 (value for n) = 12 we must - 2 to get to the number 10 5 (position in the sequence) x 3 (value for n) = 15 we must - 2 to get to the number 13 6 (position in the sequence) x 3 (value for n) = 18 we must - 2 to get to the number 16 So the nth term for our sequence is 3n-2, which means sadly Claire is incorrect.
• Question 7

Work out the nth term in this sequence:

4, 7, 10, 13, 16, 19 ...

3n+1
EDDIE SAYS
Did you find the right answer? The common difference is 3, so the first part of our term is going to be 3n. Now, we need to work out if we are adding or subtracting to/from that number? 1 (position in the sequence) x 3 (value for n) = 3. We must +1 to get to the number 4. 2 (position in the sequence) x 3 (value for n) = 6 We must +1 to get to the number 7 3 (position in the sequence) x 3 (value for n) = 9 we must +1 to get to the number 10 4 (position in the sequence) x 3 (value for n) = 12 we must +1 to get to the number 13 5 (position in the sequence) x 3 (value for n) = 15 we must +1 to get to the number 16 6 (position in the sequence) x 3 (value for n) = 18 we must +1 to get to the number 19 So, the equation we are after is 3n+1!
• Question 8

It's time for you to write the correct nth term for this sequence:

1, 6, 11, 16, 21, 26 ...

Remember, when writing don't include any spaces between your answers.  For example, 3n+1

5n-4
5n - 4
EDDIE SAYS
How did you do here? Did you remember to use the structure we've used in our previous questions? The common difference is 5, so we know this sequence begins 5n. Then we follow the usual process to work out the rest of the nth term. 1 (position in the sequence) x 5 (value for n) = 5. We must -4 to get to the number 1. 2 (position in the sequence) x 5 (value for n) = 10 We must -4 to get to the number 6 3 (position in the sequence) x 5 (value for n) = 15 we must -4 to get to the number 11 4 (position in the sequence) x 5 (value for n) = 20 we must -4 to get to the number 16 5 (position in the sequence) x 5 (value for n) = 25 we must -4 to get to the number 21 6 (position in the sequence) x 5 (value for n) = 30 we must -4 to get to the number 26 So, the answer must be 5n - 4. Well done if you got this right!
• Question 9

Work out the nth term in this sequence:

10, 17, 24, 31, 38, 45 ...

7n+3
7n + 3
EDDIE SAYS
Did you remember how to begin? Start by finding the common difference, which is 7. Let's make a note of 7n. 1 (position in the sequence) x 7 (value for n) = 7. We must +3 to get to the number 10 2 (position in the sequence) x 7 (value for n) = 14 We must +3 to get to the number 17 3 (position in the sequence) x 7 (value for n) = 21 we must +3 to get to the number 24 4 (position in the sequence) x 7 (value for n) = 28 we must +3 to get to the number 31 5 (position in the sequence) x 7 (value for n) = 35 we must +3 to get to the number 38 6 (position in the sequence) x 7 (value for n) = 42 we must +3 to get to the number 45 So there we have our answer. 7n+3 is the nth term in this sequence!
• Question 10

Work out the nth term in this sequence:

9, 13, 17, 21, 25, 29 ...

4n+5
4n + 5
EDDIE SAYS
Did you crack this last one? The common difference was 4, so we write down 4n. Then we see if we need to add or subtract to/from this number. 1 (position in the sequence) x 4 (value for n) = 4 We must +5 to get to the number 9 2 (position in the sequence) x 4 (value for n) = 8 We must +5 to get to the number 13 3 (position in the sequence) x 4 (value for n) = 12 we must +5 to get to the number 17 4 (position in the sequence) x 4 (value for n) = 16 we must +5 to get to the number 21 5 (position in the sequence) x 4 (value for n) = 20 we must +5 to get to the number 25 6 (position in the sequence) x 4 (value for n) = 24 we must +5 to get to the number 29 So, our final answer is 4n+5. That's another activity completed, well done!
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