Generate a Sequence from an Nth Term

In this worksheet, students use nth terms to find specified values in a sequence and to solve problems involving the relationships between terms in a 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

To generate a sequence from the nth term, we have to remember what an nth term actually means.

It is a position-to-term rule which allows the position within the sequence to be inputted to generate the value which corresponds to this position in the sequence.

Let's look at this in practice now to learn how we should approach finding a sequence from the nth term.

e.g. Find the first 5 terms in this sequence: 4n - 6

For this, we have to substitute the values n = 1, n = 2, n = 3, n = 4 and n = 5 into the nth term:

When n = 1, 4n - 6 = (4 × 1) - 6 = -2

When n = 2, 4n - 6 = (4 × 2) - 6 = 2

When n = 3, 4n - 6 = (4 × 3) - 6 = 6

When n = 4, 4n - 6 = (4 × 4) - 6 = 10

When n = 5, 4n - 6 = (4 × 5) - 6 = 14

So our first five terms for the sequence 4n - 6 are: -2, 2, 6, 10, 14.

Let's look at another example now which asks us to apply our knowledge of the nth term to solve a problem.

e.g. Find the difference between the 100th and the 90th terms in this sequence: 5n + 2

The approach to this is very similar to the last question, but we use n = 90 and n = 100 instead:

When n = 100, 5n + 2 = (5 × 100) +2 = 502

When n = 90, 5n + 2 = (5 × 90) +2 = 452

We can then find the difference between these two terms by subtracting these two values:

502 - 450 = 50

So the difference between the 90th and 100th terms in the sequence 5n + 2 is 50

In this activity, we will use nth terms to find specified values in a sequence and to solve problems involving the relationships between terms in a sequence.

Complete the sentence below summarising how we use the nth term to find the value of any term in a sequence.

Try to use the same word which features in the Introduction so that you are marked correctly.

What are the first five terms of this sequence?

2n - 3

What are the first five terms of this sequence?

3n + 2

What is the 100th term of this sequence?

3n - 5

James thinks that the 4th term of the sequence 3n - 1 is one third of the 4th term of the sequence 7n + 5.

Is James correct?

Yes

No

What is the difference between the 100th and 50th terms of this sequence?

5n - 1

Match each sequence below with its correct first 5 terms.

Column B

2n + 1
5, 9, 13, 17
3n - 2
-1, 5, 11, 17, 23
4n + 1
3, 5, 7, 9, 11
6n - 7
1, 4, 7, 10, 13

Which is the first number to appear in both the sequences described by 2n + 1 and 4n + 1

What is the average of the first 5 terms of this sequence?

7n - 3

Which is larger?

The 12th term of 3n + 2 or the 6th term of 6n + 1.

12th term of 3n + 2

6th term of 6n + 1

They're the same

• Question 1

Complete the sentence below summarising how we use the nth term to find the value of any term in a sequence.

Try to use the same word which features in the Introduction so that you are marked correctly.

EDDIE SAYS
It's important to use the correct vocabulary in relation to sequences. When we change a letter into a number in algebra, this is called substitution so we need to substitute the relevant value of n (i.e. the position in the sequence) into the nth term. Remember this important fact to support you in the rest of this activity.
• Question 2

What are the first five terms of this sequence?

2n - 3

EDDIE SAYS
Here we are going to substitute the numbers 1 to 5 in for n in the nth term. When n = 1, 2n - 3 = (2 × 1) - 3 = -1 When n = 2, 2n - 3 = (2 × 2) - 3 = 1 When n = 3, 2n - 3 = (2 × 3) - 3 = 3 When n = 4, 2n - 3 = (2 × 4) - 3 = 5 When n = 5, 2n - 3 = (2 × 5) - 3 = 7 Did you type those values accurately into the gaps in the sequence and in the correct order?
• Question 3

What are the first five terms of this sequence?

3n + 2

EDDIE SAYS
Again, we need to substitute the numbers 1 to 5 in for n in the nth term to answer this. When n = 1, 3n + 2 = (3 × 1) + 2 = 5 When n = 2, 3n + 2 = (3 × 2) + 2 = 8 When n = 3, 3n + 2 = (3 × 3) + 2 = 11 When n = 4, 3n + 2 = (3 × 4) + 2 = 14 When n = 5, 3n + 2 = (3 × 5) + 2 = 17 How did you get on there?
• Question 4

What is the 100th term of this sequence?

3n - 5

295
EDDIE SAYS
This is the same as before, just using a bigger number. All we have to do here is substitute n = 100 into the nth term: When n = 100, 3n - 5 = (3 × 100) - 5 = 295 That would have been very time-consuming to find using a term-to-term rule, wouldn't it?
• Question 5

James thinks that the 4th term of the sequence 3n - 1 is one third of the 4th term of the sequence 7n + 5.

Is James correct?

Yes
EDDIE SAYS
Lets split this question up. The 4th term of 3n - 1 is (3 × 4) - 1 = 11 The 4th term of 7n + 5 is (7 × 4) + 5 = 33 Is 11 one third of 33? It is! So James is correct.
• Question 6

What is the difference between the 100th and 50th terms of this sequence?

5n - 1

250
EDDIE SAYS
Did you spot that this is like one of the examples in the Intro? We need to find the 100th term and the 50th term and then subtract them. When n = 100, 5n - 1 = (5 × 100) - 1 = 499 When n = 50, 5n - 1 = (5 × 50) - 1 = 249 If we subtract the 50th term from the 100th, we reach: 499 - 249 = 250
• Question 7

Match each sequence below with its correct first 5 terms.

Column B

2n + 1
3, 5, 7, 9, 11
3n - 2
1, 4, 7, 10, 13
4n + 1
5, 9, 13, 17
6n - 7
-1, 5, 11, 17, 23
EDDIE SAYS
We can work these out by substituting the values of n from 1-5 into each nth term, but do we even need to work these all out? We can also look for patterns in the numbers to help us match these pairs quicker. 2n + 1 must go up in 2s, so which sequence does this match with? 3n - 2 will match a sequence with gaps of 3. 4n + 1 will match a sequence with gaps of 4. 6n - 7 will match a sequence with gaps of 6. Did these observations help you match these more quickly?
• Question 8

Which is the first number to appear in both the sequences described by 2n + 1 and 4n + 1

5
EDDIE SAYS
The only way we find this out is to work out the sequences up to the 10th term. 2n + 1: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 4n + 1: 5, 9, 13, 17, 21, 25, 29, 33, 37, 41 Which numbers appear in both? 5, 9, 13, 17, 21, etc. What is the first number which appears in both? It's 5.
• Question 9

What is the average of the first 5 terms of this sequence?

7n - 3

18
EDDIE SAYS
Our first step here is to work out the first five terms of this sequence by substituting n = 1, n = 2, etc. into the nth term. 7n - 3 = 4, 11, 18, 25, 32, ... If we see the word 'average', we need to add all the values together and divide by the number of values present. 4 + 11 + 18 + 25 + 32 = 90 90 ÷ 5 = 18
• Question 10

Which is larger?

The 12th term of 3n + 2 or the 6th term of 6n + 1.

12th term of 3n + 2
EDDIE SAYS
This looks harder than it actually is. All we need to do is to work out these two values in the way we have before and then compare them. 12th term of 3n + 2 = (3 × 12) + 2 = 38 6th term of 6n + 1 = (6 × 6) + 1 = 37 Which of these values is larger? Congratulations on completing this activity! Why not practise some more activities on the nth term now that you are on a roll?
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