Find a rule for any term of a sequence that goes up or down by the same amount, and test whether a number belongs to it. It is on Foundation and Higher, aimed at grade 4. Free questions with answers are on this page.
A linear sequence goes up (or down) by the same amount each time. That common difference is the coefficient of n in the nth term. Sequence 5,8,11,14,… has difference 3, so the nth term is 3n+k. The zeroth term (step back once from the first term) is k: here 5−3=2, so 3n+2. Check at n=1 and n=2.
Is 80 in the sequence? Solve 3n+2=80 and ask whether n is a positive integer. Finding a term number is substituting into the nth term. Quadratic sequences (second difference constant) are a later Higher skill.
For an arithmetic sequence the nth term is an+b where a is the common difference and b is the zero term (first term minus a). To test membership, solve an+b= the number and check n is a positive integer.
Objectives
In a linear sequence the difference between terms is constant.
The nth term is (difference) × n, then add or subtract a constant.
Put n = 1, 2 and 3 back into your rule.
Key points
Find the common difference of a linear (arithmetic) sequence.
Write the nth term as dn+c where d is the difference and c is the 'zero term'.
Handle decreasing sequences, where the difference is negative.
Use the nth term to find any term, e.g. the 50th.
Worked examples
Example 1
A sequence starts at 3 and increases by 2 each time. What is term 4?
2 marks
The first term is 3 and each term adds 2.
Term 4 = 3 + (4 − 1) × 2 = 3 + 6.
Term 4 is 9.
Answer: 9
Common slips
Slip: The first term is 5 and each term adds 2. That changes a number from the question, so it answers a different question.
Slip: Term 7 = 3 + (4 − 1) × 2 = 3 + 6. That changes a number in the working, so the step no longer follows from the one before.
Slip: Term 8 is 9. That does not match the working above it.
Example 2
The nth term of a sequence is 3n + 2. Work out the 7th term.
2 marks
Put n = 7 into 3n + 2.
3 × 7 = 21.
21 + 2 = 23.
Answer: 23
Common slips
Slip: Put n = 9 into 3n + 2. That changes a number from the question, so it answers a different question.
Slip: 3 + 7 = 10. That uses addition, but this step needs multiplication.
Slip: 21 − 2 = 19. That uses subtraction, but this step needs addition.
Example 3
Find the nth term of the sequence 7, 11, 15, 19, ...
2 marks
The terms go up by 4 each time, so the rule starts 4n.
When n = 1, 4n gives 4. The first term is 7, so add 3.
The nth term is 4n+3.
Answer: 4n+3
Common slips
Slip: The terms go up by 6 each time, so the rule starts 4n. That changes a number from the question, so it answers a different question.
Slip: When n = 4, 4n gives 4. The first term is 7, so add 3. That changes a number in the working, so the step no longer follows from the one before.
Slip: The nth term is 8n+3. That does not match the working above it.
Questions and answers
GCSE Maths sequences (nth term) questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
The sequence starts 3, 7, 11, 15. What is the next term?
Show answer
Answer: 19
The difference is 4.
Add 4 to the last term.
M1 — The difference is 4.
A1 — Answer 19
Question 2 Easy2 marks
The nth term is 2n + 1. Find the 6th term.
Show answer
Answer: 13
Substitute n = 6.
2 × 6 + 1.
M1 — Substitute n = 6.
A1 — Answer 13
Question 3 Medium2 marks
The nth term is 5n + 3. Is 23 in the sequence? Answer yes or no.
Show answer
Answer: yes
Solve 5n + 3 = 23.
n has to be a positive integer.
M1 — Solve 5n + 3 = 23.
A1 — Answer yes
Question 4 Medium2 marks
The nth term is 3n + 2. Which term is 14?
Show answer
Answer: 4
Solve 3n + 2 = the term.
n is the position.
M1 — Solve 3n + 2 = the term.
A1 — Answer 4
Question 5 Hard2 marks
The arithmetic sequence 4, 7, 10 has common difference 3. Find the 20th term.
Show answer
Answer: 61
Nineteen steps of 3 come after the first term.
4 + 57.
M1 — Nineteen steps of 3 come after the first term.
A1 — Answer 61
Question 6 Hard2 marks
The nth term is 2n − 3. The term is 7. What is n?
Show answer
Answer: 5
Add 3, then divide by the coefficient of n.
The value is 5.
M1 — Add 3, then divide by the coefficient of n.
A1 — Answer 5
Question 7 Exam-style3 marks
The nth term is 3n − 3. The term is 12. What is n?
Show answer
Answer: 5
Add 3, then divide by the coefficient of n.
The value is 5.
M1 — Add 3, then divide by the coefficient of n.
M1 — The value is 5.
A1 — Answer 5
Common mistakes
Most lost marks on this topic are one slip.
Writing the difference as the nth term.
4n gives 4, 8, 12, not 3, 7, 11. You still need the adjustment.
Using n = 0 for the first term.
The first term is n = 1.
Adding the difference to n instead of multiplying.
The difference multiplies n.
Spot the slip
The difference of 3, 7, 11, 15 is
4
3
15
Show the slip
4 Each term is 4 more than the one before.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
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.
Always check your rule with n=1; it takes five seconds and catches most slips.
For 'is it a term', the mark is for showing n is (or is not) a whole number — write that sentence.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.