GCSE MATHS • HIGHER TIER
Nth Term Sequences Made Simple
Describe arithmetic sequences algebraically, form nth-term rules and calculate any required term.
Clear method
Exam practice
Worked proofs
Nth Term Sequences in Four Moves
1
Find the Difference
Find the constant difference between consecutive terms.
2
Use n
Use the common difference as the coefficient of n.
3
Adjust
Compare the generated sequence with the original and add or subtract a constant.
4
Substitute
Put the required term position into the rule to calculate a specific term.
What You’ll Learn
Understand what an nth-term formula represents.
Find nth-term rules for arithmetic sequences.
Use the common difference as the coefficient of n.
Adjust a rule using a constant.
Find specific terms by substitution.
Work with decreasing sequences.
Test whether a number appears in a sequence.
Key Rules
Common Difference
Find the difference between consecutive terms. For an arithmetic sequence, this difference stays the same.
Coefficient of n
Use the common difference as the number in front of n. Example: Difference = 3 → start with 3n.
Adjust the Constant
Compare the n sequence with the original sequence, then add or subtract a constant to make the terms match.
Find Any Term
Once you know the nth term formula, substitute the term position for n. Example: if the rule is 4n + 2, the 20th term is 4(20) + 2 = 82.
Worked Proof: Nth Term Sequences
Step-by-Step Method
- 1. Find the common difference.
- 2. Use it as the coefficient of n.
- 3. Compare the generated sequence with the original.
- 4. Add or subtract a constant.
- 5. Write the nth term rule.
More Worked Examples
1
Using the nth Term to Find Specific Terms
Once the nth term is known, substitute the position number into the formula.
The nth term is 4n + 2. Find the 20th term.
2
Negative and Decreasing Sequences
Some sequences decrease.
20, 17, 14, 11, 8
Difference = -3
Difference = -3
3
Finding Whether a Number Appears in a Sequence
The nth term can be used to determine whether a specific number belongs to a sequence.
Sequence rule: 4n + 1
Does 101 appear?
4n + 1 = 101
4n = 100
n = 25
Does 101 appear?
4n + 1 = 101
4n = 100
n = 25
4
Visual nth Term Table
n = 1 → 5
n = 2 → 8
n = 3 → 11
n = 4 → 14
= 10π/3 cm
n = 3 → 11
n = 4 → 14
= 10π/3 cm
- Common GCSE Mistakes
- Using the wrong common difference.
- Forgetting to adjust the constant.
- Making arithmetic mistakes.
- Confusing term position with term value.
- Making errors when substituting into nth-term formulas.
- Exam Tips
- Find the common difference first.
- Use the difference as the coefficient of n.
- Compare your generated sequence carefully.
- Adjust the constant only after comparing terms.
- Substitute the position number, not the term value.
- Check your rule against more than one term.
How to Improve at This Topic
Begin with simple increasing arithmetic sequences.
Move on to decreasing sequences once confident.
Create tables and compare generated sequences with the original.
Write your own sequences and derive their nth-term rules.
Practise exam-style questions regularly to improve speed and accuracy.
Ready to Practise?
Printable Worksheet
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Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Nth Term Sequences Tutorial
Step-by-step walkthroughs of Nth Term Sequences with clear methods and exam tips.
Frequently Asked Questions
It is a formula used to find any term in a sequence, where n represents the term position.
Find the common difference and use it as the coefficient of n.
Substitute the required position number into the nth-term formula.
Yes. A decreasing arithmetic sequence can have a negative coefficient of n.
Set the nth-term formula equal to that number and solve for n.
Arithmetic Sequences, Geometric Sequences, Algebra, Substitution and Graphs.