A sequence is an ordered list, so position matters. A term-to-term rule tells you how to get from one term to the next. A position-to-term rule lets you calculate any term directly from its position n. In a linear sequence, the difference between consecutive terms is constant. This difference becomes the coefficient of n in the nth-term rule. Compare the sequence with that times table to find the extra constant. Use n = 1 to check the first term, then test another position before using the rule for a distant term.
Learning objectives
- Find and use a linear sequence rule.
- Decide whether a value belongs to a sequence.
Key ideas
A linear sequence has a constant first difference.
For first term and common difference, use this position-to-term rule.
Term positions start at one unless the question states otherwise.
Check a proposed rule using the first few positions.
A sequence member must correspond to a positive whole-number position.
Worked examples
Example 1
Find the position-to-term rule and the twentieth term.
Answer
Example 2
Test whether the proposed value is a sequence member.
This is not a whole-number position.
Answer
Question styles to prepare for
Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.
AO1 · Methods
Continue a sequence or find and use a linear nth-term rule.
AO2 · Reasoning
Decide whether a number belongs to a sequence, giving a reason.
AO3 · Problem solving
Translate a growing diagram pattern into a rule and use it for a later stage.
When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.
Avoid this mistake
The common difference is only the coefficient of the position. Check the constant adjustment using the first term.
Try it yourself
Attempt each question before opening its answer.
Check 1: Find the next two terms of 20, 17, 14, ...
Answer: 11, 8
Subtract 3 each time.
Check 2: Is 46 in the sequence with nth term 6n + 2?
Answer: No
6n + 2 = 46 gives n = 44/6 = 22/3. This is not a whole-number position.
Explain it in your own words
- Explain the difference between adding 3 each time and using 3n.
- Derive an nth-term rule and check n = 1 and n = 4.
- Create a decreasing linear sequence and state its nth term.
Keep practising
Download the compact topic guide (PDF) · Practise this topic in Blitz · All foundation notes
Specification source
AQA Algebra specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.