 # Use the Nth Term in a Sequence

In this worksheet, students will use the nth term to confirm if a number does or does not appear in a given 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

One of the most common questions you will see when we are working with sequences is:

"Is the number X in this sequence?"

There are a couple of ways we can work this out:

Method 1: The easy spot

e.g. Is the number 275 in this sequence: 4, 8, 12, 16, 20, ...?

Straight away we can spot that 275 seems wrong, as all the numbers in this sequence are even.

e.g. Is the number 97 in this sequence: 3, 8, 13,18, 23, ...?

This one is a bit trickier.

Did you notice that all the numbers in the sequence end in 3 or 8?

This means that 97 can't be in this sequence.

Method 2: Using the nth term

To use this, we have to remember that the 'n' in the nth term refers to the position.

Therefore, it is never possible to a number in the 1.5th position, only whole numbers.

e.g. Is 75 in this sequence: 2, 5, 8, 11, ...?

Step 1: Find the nth term of the sequence.

For this sequence, our nth term would be 3n - 1.

Step 2: Make this nth term equal to the number you are looking for and then solve the equation.

3n - 1 = 75

3n = 76

n = 25.3333

Because n isn't a whole number, this means that 75 cannot be in this sequence.

e.g. Is 224 in this sequence: 8, 14, 20, 26, ...?

Step 1: Find the nth term of the sequence.

For this sequence, our nth term would be 6n + 2.

Step 2: Make this nth term equal to the number you are looking for and then solve the equation.

6n + 2 = 224

6n = 222

n = 37

Because n is a whole number, 224 is in this sequence.

In this activity, we will use the nth term to confirm if a number does or does not appear in a given sequence. If we can, we will use the 'easy spot' method first in order to save time and effort.

Complete the sentence below to summarise which values of n indicate that a value does appear in a sequence.

38 is in the sequence 3n + 2.

What position is it in?

43 is in the sequence 5n - 7.

What position is it in?

Is 24 in this sequence: 3, 8, 13, 18, ...?

Yes

No

Is 273 in this sequence: 2, 6, 10, 14, ...?

Yes

No

Does 33 appear in the sequence 2n + 5?

Yes

No

Match each nth term below with the correct number that appears in its sequence.

## Column B

4n + 1
29
3n - 7
201
2n + 3
153

For each of the nth terms below, select if the number in the accompanying bracket does (Y) or does not (N) appear in the given sequence.

Which of the numbers listed below appear in the sequence 3n - 1?

8

17

21

26

32

40

Complete the sentence below to summarise our method for confirming if a number does or does not appear in a given sequence.

8

17

21

26

32

40

• Question 1

Complete the sentence below to summarise which values of n indicate that a value does appear in a sequence.

EDDIE SAYS
Remember that 'n' refers to a number's position in a sequence. For this reason, it is impossible to have a position (or value of n) which is not a whole number. The formal name for a whole number or a number which is not a fraction is an integer.
• Question 2

38 is in the sequence 3n + 2.

What position is it in?

12
12th
twelfth
Twelfth
EDDIE SAYS
Let's start by setting this nth term up as an equation with 38 as our answer: 3n + 2 = 38 Now all we need to do is to solve this to find the value of n where this equation is true: 3n + 2 - 2 = 38 - 2 3n ÷ 3 = 36 ÷ 3 n = 12 So the number 38 will appear in the 12th position in this sequence.
• Question 3

43 is in the sequence 5n - 7.

What position is it in?

10
10th
tenth
Tenth
EDDIE SAYS
Let's start by setting this nth term up as an equation with 43 as our answer: 5n - 7 = 43 Now all we need to do is to solve this to find the value of n where this equation is true: 5n - 7 + 7 = 43 + 7 5n ÷ 5 = 50 ÷ 5 n = 10 So the number 43 will appear in the 10th position in this sequence.
• Question 4

Is 24 in this sequence: 3, 8, 13, 18, ...?

No
EDDIE SAYS
All the numbers in this sequence end in 3 or 8, so 24 cannot be in this sequence. Did you make this 'easy spot' here?
• Question 5

Is 273 in this sequence: 2, 6, 10, 14, ...?

No
EDDIE SAYS
All the numbers in this sequence are even and 273 is odd, so it cannot be in this sequence. We can always look for ending numbers, multiples and odd/even differences in order to make an 'easy spot' with sequences.
• Question 6

Does 33 appear in the sequence 2n + 5?

Yes
EDDIE SAYS
We should start by setting up this up as an equation with the target number as our answer: 2n + 5 = 33 We need to find out if we have a whole number value for n here, which would indicate that this value does appear in this sequence. Let's solve this equation now to find the value of n: 2n + 5 - 5 = 33 - 5 2n ÷ 2 = 28 ÷ 2 n = 14 As n is an integer, this means that 33 does appear in this sequence.
• Question 7

Match each nth term below with the correct number that appears in its sequence.

## Column B

4n + 1
201
3n - 7
29
2n + 3
153
EDDIE SAYS
It's a bit of a process of elimination here. We can test each value in each option by creating equations with these values as answers. It's a good idea to start with the ones that look similar. 4n + 1 and 201 both end in a 1 so let's try these first: 4n + 1 = 201 4n + 1 - 1 = 201 - 1 4n ÷ 4 = 200 ÷ 4 n = 50 As this value is a whole number, we know that 201 does appear in the sequence 4n + 1. Can you follow this process to test the other two options independently and find the correct matches?
• Question 8

For each of the nth terms below, select if the number in the accompanying bracket does (Y) or does not (N) appear in the given sequence.

EDDIE SAYS
For each of these options, we need to set up an equation with the accompanying number as the answer, solve them and see if we get an answer which is an integer. 2n + 1 = 10 2n + 1 - 1 = 10 - 1 2n ÷ 2 = 9 ÷ 2 n = 4.5 As this is not a whole number, this value does not appear in this sequence so make sure you have selected 'No' here. Can you follow this same process to test the other two options independently?
• Question 9

Which of the numbers listed below appear in the sequence 3n - 1?

8
17
26
32
EDDIE SAYS
We could use the nth term method here, but it's definitely easier just to write out the whole sequence until it passes these values. We calculate each value in the sequence by substituting the relevant values for n (i.e. n = 1, n = 2, n = 3, etc.) into the nth term. Our sequence here is: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, ... Our highest number is 40, so let's stop now we have passed this. Does this make it easier to spot which of these options appear?
• Question 10

Complete the sentence below to summarise our method for confirming if a number does or does not appear in a given sequence.

EDDIE SAYS
In order to confirm if a number appears in a given sequence or not, we need to create an equation with this number as the answer and solve it to find the value of n. If the value we reach is an integer (whole number), then this means that the number does appear in the target sequence. If the value we reach is not an integer, then this means the opposite and that the number does not appear in the target sequence. Great job on this activity! Are you ready to try another sequences activity now?
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