Continue a sequence from a term-to-term rule or a position rule, and know the square, cube and triangular numbers. It is a Foundation topic, aimed at grade 2. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Objectives
Use a term-to-term rule to find the next terms.
Use a position-to-term rule to find a term without listing the rest.
Recognise square, cube, triangular and Fibonacci-type sequences.
Key points
Use a term-to-term rule such as add 4 or multiply by 2 to find the next terms.
Use a position-to-term rule to find any term without listing the ones before it.
Know the square, cube and triangular numbers and Fibonacci-type sequences.
Find the rule by looking at the gaps between neighbouring terms.
Worked examples
Example 1
The first term of a sequence is 5 and the rule is add 3. Write the first four terms.
2 marks
Start with the first term, 5
Add 3 each time: 5 + 3 = 8
Keep going: 8 + 3 = 11 and 11 + 3 = 14, so the terms are 5, 8, 11, 14
Answer: 5, 8, 11, 14
Common slips
Slip: Start with 3 The first term is given as 5.
Slip: 5 × 3 = 15 The rule is add 3, not multiply by 3.
Slip: 5, 8, 11, 13 11 + 3 is 14, so check each addition.
Example 2
Here is a sequence: 2, 3, 5, 8, 13. Each term is the sum of the two terms before it. Write the next two terms.
2 marks
Check the rule: 2 + 3 = 5 and 3 + 5 = 8
The next term is 8 + 13 = 21
The term after that is 13 + 21 = 34
Answer: 21, 34
Common slips
Slip: The gaps are 1, 2, 3, 5 so add 5 The gaps are not constant, so there is no single number to add.
Slip: The next term is 13 + 5 = 18 Add the last two terms, not the last term and a gap.
Slip: The term after that is 21 + 8 = 29 Use the two terms just before the one you want.
Example 3
A sequence starts at 3. The rule is multiply by 2 then subtract 1. Work out the 4th term.
3 marks
Apply the rule to 3: 3 × 2 - 1 = 5
Apply it to 5: 5 × 2 - 1 = 9
Apply it to 9: 9 × 2 - 1 = 17, so the 4th term is 17
Answer: 17
Common slips
Slip: 3 × 2 = 6 The rule has two parts, multiply then subtract 1.
Slip: 5 × 2 - 1 = 10 - 1 = 8 10 - 1 is 9, so recheck the subtraction.
Slip: The 4th term is 9 because it is the 3rd answer The first term is given, so the 4th term needs three applications.
Questions and answers
GCSE Maths sequences: rules and special sequences questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
The first term of a sequence is 7 and the rule is add 4. What is the 4th term?
Show answer
Answer: 19
Start with 7.
7 + 4 = 11.
11 + 4 = 15.
15 + 4 = 19.
M1 — Start with 7.
A1 — Answer 19
Question 2 Easy2 marks
The first term of a sequence is 10 and the rule is add 5. What is the 5th term?
Show answer
Answer: 30
Start with 10.
10 + 5 = 15.
15 + 5 = 20.
20 + 5 = 25.
25 + 5 = 30.
M1 — Start with 10.
A1 — Answer 30
Question 3 Medium2 marks
What is the next term in 5, 11, 17, 23, ...?
Show answer
Answer: 29
The gap each time is 6.
23 + 6 = 29.
M1 — The gap each time is 6.
A1 — Answer 29
Question 4 Medium2 marks
What is the 7th square number?
Show answer
Answer: 49
The 7th square number is 7 × 7.
7 × 7 = 49.
M1 — The 7th square number is 7 × 7.
A1 — Answer 49
Question 5 Hard2 marks
a sequence starts at 1. The rule is multiply by 2 then add 1. Work out the 5th term.
Show answer
Answer: 31
1 × 2 + 1 = 3.
3 × 2 + 1 = 7.
7 × 2 + 1 = 15.
15 × 2 + 1 = 31.
M1 — 1 × 2 + 1 = 3.
A1 — Answer 31
Question 6 Hard2 marks
The rule for the sequence is multiply the position by 5 then subtract 2. What is the 8th term?
Show answer
Answer: 38
Use the position 8: 8 × 5 = 40.
40 - 2 = 38.
M1 — Use the position 8: 8 × 5 = 40.
A1 — Answer 38
Question 7 Exam-style3 marks
The first four terms of a sequence are 1, 3, 6, 10. Write down the next two terms.
Show answer
Answer: 15, 21
The gaps are 2, 3, 4, so the next gaps are 5 and 6.
10 + 5 = 15 and 15 + 6 = 21.
M1 — Sees gaps 2, 3, 4 or gaps increasing by 1
A1 — 15
A1 — 21
Common mistakes
Most lost marks on this topic are one slip.
Applying the rule one time too many
To reach the 4th term from the 1st term you apply the rule three times
Using the first term as the gap
The gap is found by subtracting neighbouring terms
Adding the gap when the numbers are going down
A falling sequence uses subtract
Spot the slip
A sequence starts 4, 7, 10, 13. Ada says the next term is 15. What did she do?
She added 2 instead of the gap of 3
She added 4 because the first term is 4
She doubled the last term
Show the slip
She added 2 instead of the gap of 3 The gap is 3 each time, so the next term is 13 + 3 = 16.
Exam tips
This topic can appear on a calculator or a non-calculator paper. Show enough working for a method mark.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.