# Understand Linear Sequences (term-to-term rules)

In this worksheet, students will find term-to-term rules for linear sequences and apply these to find next or specified terms.

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

A linear sequence has a difference between the terms that is the same each time, so it increases or decreases by equal steps.

This means we can continue such a sequence using a term-to-term rule.

e.g. Find the 10th number in this sequence:

2, 6, 10, 14, ...

If we look at the differences between the terms, each term is 4 more than the previous.

So the term-to-term rule is + 4.

If we continue this sequence, it would become:

2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, etc...

The tenth term is 38.

In this activity, we will find term-to-term rules for linear sequences and apply these to find next or specified terms.

You should not use a calculator for this activity, but rather practise your mental arithmetic skills.

Complete the sentence below to define a linear sequence.

What is the rule for this sequence?

6, 11, 16, 21, ...

Your first blank below should contain either a + or - sign, and your second should contain a number.

What is the 11th term in this sequence?

3, 7, 11, 15, ...

James predicts that the 20th number in the sequence 2, 5, 8, 11, ... will be double the 10th.

Is he correct?

Yes

No

What is the first number in the sequence 1, 5, 9, ... which will be greater than 50?

How many positive numbers are in this sequence?

25, 23, 21, ...

Match each sequence below with its correct next two terms.

## Column B

12, 15, 18, ...
21, 24
1, 2, 3, 4, ...
5, 6
20, 15, 10, 5, ...
0, -5
16, 11, 6, ...
1, -4

Match each sequence below with its correct 12th term.

## Column B

3, 5, 7, 9, ...
25
10, 13, 16, 19, ...
1
11, 8, 5, 2, ...
-22
1, 1, 1, 1, ...
43

Consider this sequence:

3, 8, 13, 18, ...

Which of the numbers below will appear in this sequence?

23

38

0

-3

25

10000457893

Look at this final sequence:

22, 18, 14, ...

Which of the numbers below will appear in this sequence?

4

-6

-5

0

26

10

• Question 1

Complete the sentence below to define a linear sequence.

EDDIE SAYS
A linear sequence always increases or decreases by the exact same amount. The correct phrasing to describe this easily is to say that such a sequence has a common difference. Now you know this term, be sure to keep your eyes open for it in questions or working to help you spot a linear sequence.
• Question 2

What is the rule for this sequence?

6, 11, 16, 21, ...

Your first blank below should contain either a + or - sign, and your second should contain a number.

EDDIE SAYS
The easiest way to find the rule is to subtract one term from the number following it (or subtract adjacent terms). You can choose any terms you would like to calculate this with, although it makes sense to choose the easiest terms unless you want an added challenge! e.g. 21 - 16 = 5 So we think our rule is + 5. Let's check this rule in one more pair to be sure: 11 + 5 = 16 So we can be totally confident that this rule is definitely correct.
• Question 3

What is the 11th term in this sequence?

3, 7, 11, 15, ...

43
EDDIE SAYS
All we have to do here is find the common difference and keep creating the sequence until we have 11 numbers. To find our common difference, let's subtract two adjacent terms: 15 - 11 = 4 --> So our rule is + 4 Then let's keep creating the terms using the previous term: 5th: 15 + 4 = 19 6th: 19 + 4 = 23 7th: 23 + 4 = 27 8th: 27 + 4 = 31 9th: 31 + 4 = 35 10th: 35 + 4 = 39 11th: 39 + 4 = 43
• Question 4

James predicts that the 20th number in the sequence 2, 5, 8, 11, ... will be double the 10th.

Is he correct?

No
EDDIE SAYS
James has made a really common mistake that often occurs when students work with sequences. We cannot assume any relationship between terms which are not adjacent (i.e. double 10 is 20 so let's double the answer to the 10th term to find the 20th). The only way to work these terms out is to find the common difference and keep applying it until we have 20 numbers in the sequence. Let's find the 10th and 20th terms and compare them: The 10th term is 29. The 20th term is 59. So this information proves that James' theory will definitely not work.
• Question 5

What is the first number in the sequence 1, 5, 9, ... which will be greater than 50?

53
EDDIE SAYS
Here we can find our rule, and keep applying it until we reach the first term which passes above 50. The rule here is: 5 - 1 = 4 --> So our rule is + 4 If we continue the sequence we will reach: 49, 53, 57, ... We can see that the first number in the sequence which is bigger than 50 is 53. Did you find that?
• Question 6

How many positive numbers are in this sequence?

25, 23, 21, ...

13
EDDIE SAYS
Here our sequence is decreasing, so we know that we need to use a - sign in our rule. 25 - 23 = 2 --> So our rule is - 2 If we continue this sequence, we will eventually drop below zero. We can see from this list how many numbers are in the sequence before that happens: 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1, -1, -3, ... There are 13 positive numbers in this list before the sequence becomes negative.
• Question 7

Match each sequence below with its correct next two terms.

## Column B

12, 15, 18, ...
21, 24
1, 2, 3, 4, ...
5, 6
20, 15, 10, 5, ...
0, -5
16, 11, 6, ...
1, -4
EDDIE SAYS
We need to find the term-to-term rule in each sequence, then use it to find the next two terms in each case. Let's look at the final option together as an example: 16, 11, 6, ... To find our common difference, we need to subtract two adjacent terms: 16 - 11 = 5 --> So our rule is - 5 Let's find our next two terms using the previous term to subtract 5 from: 4th: 6 - 5 = 1 5th: 1 - 5 = -4 Can you use this example to match the other sequences and terms independently?
• Question 8

Match each sequence below with its correct 12th term.

## Column B

3, 5, 7, 9, ...
25
10, 13, 16, 19, ...
43
11, 8, 5, 2, ...
-22
1, 1, 1, 1, ...
1
EDDIE SAYS
Here we need to continue each sequence until there are 12 numbers present. Let's look at the first sequence together: 3, 5, 7, 9, ... Our rule here is: 5 - 3 = 2 --> So our rule is + 2 Then let's apply this rule to work up to 12 terms in the sequence: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, .... Can you follow this example to match the other sequences and terms independently? Did you notice that the final sequence is not changing at all? So our rule here would be + 0, which is pretty easy to apply, isn't it?
• Question 9

Consider this sequence:

3, 8, 13, 18, ...

Which of the numbers below will appear in this sequence?

23
38
10000457893
EDDIE SAYS
If we look at the sequence, we can see that every number has either a 3 or an 8 at the end of it. Our rule here is + 5, and if we think about our 5 times table, all the answers there end in either a 0 or a 5. Therefore, this is quite a unique common difference where we will see a repeating pattern in our final digits. Sequences involving 10's will also operate in a similar way but, in this case, the ending digits will always stay exactly the same. Sequences which use a '+' in their rules can't go down, so all the negative number options can also be ruled out.
• Question 10

Look at this final sequence:

22, 18, 14, ...

Which of the numbers below will appear in this sequence?

4
-6
10
EDDIE SAYS
This sequence does not have a neat and tidy pattern to spot as with the previous question. We need to continue this sequence by subtracting 4 each time, to spot the numbers which do appear in the list. As this sequence is decreasing, the terms will pass zero and become negative at some point. Great job completing this activity! If you have time and are feeling motivated, why not move on now to learn about the nth term in sequences?
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