Finding the nth term of a sequence is a core skill in the GCSE Maths curriculum, appearing on all major exam boards like AQA, Edexcel, and OCR. It's a gateway to more advanced algebra and pattern spotting. This guide will walk you through the process for linear sequences (where the difference between terms is constant), breaking it down into simple, manageable steps.
What is the nth Term?
The nth term is simply a formula that allows you to calculate any term in a sequence, without having to write out all the previous ones. If you want the 10th term, you plug n = 10 into the formula. If you want the 100th term, you plug in n = 100. It’s a powerful tool!
For a linear sequence, the nth term formula is always in the form:
an + b
Where:
a is the common difference between terms.
b is a constant we need to find.
nis the term position (st, 2nd, 3rd...).
The Step-by-Step Method
Follow this foolproof, three-step method every time.
flowchart TD
A["Start: Sequence e.g. 5, 8, 11, 14..."] --> B["Step 1: Find the common difference, Subtract consecutive terms"]
B --> C["Step 2: Write dn, Common difference x n e.g. 3n"]
C --> D["Step 3: Compare to original sequence, Find the adjustment constant"]
D --> E["Step 4: Write final formula, e.g. nth term = 3n + 2"]
E --> F["Check: Plug in n=1,2,3 to verify"]
Step 1: Find the Common Difference (a)
Look at the sequence. Subtract the first term from the second term (or any term from the one after it).
Example: For the sequence 5, 8, 11, 14, ...
8 - 5 = 3
11 - 8 = 3
The common difference is 3. So the first part of our formula is 3n.
Step 2: Compare an to the Original Sequence
Now, create a mini-table. Write what 3n gives you for n=1, 2, 3..., and compare it to the actual sequence.
| Term Position (n) | 3n (Our guess) | Actual Sequence | Difference (Actual - 3n) |
| :--- | :--- | :--- | :--- |
| 1 | 3 x 1 = 3 | 5 | 5 - 3 = 2 |
| 2 | 3 x 2 = 6 | 8 | 8 - 6 = 2 |
| 3 | 3 x 3 = 9 | 11 | 11 - 9 = 2 |
Notice the difference is always +2. This is our constant b.
Step 3: Write the Final Formula
Combine the common difference (a=3) and the constant (b=2).
nth term = 3n + 2
Step 4: Check Your Answer!
Always check by substituting n=1, 2, 3 into your formula to see if you get the original sequence.
n=1: (3 x 1) + 2 = 5 ✅
n=2: (3 x 2) + 2 = 8 ✅
n=3: (3 x 3) + 2 = 11 ✅
Worked Example: Slightly Trickier
Find the nth term of: 10, 7, 4, 1, -2...
Common Difference: 7 - 10 = -3. The sequence is decreasing by 3 each time. So a = -3. First part: -3n.
Compare:
n=1: -3 x 1 = -3. Actual term is 10. Difference: 10 - (-3) = 13.
n=2: -3 x 2 = -6. Actual term is 7. Difference:7 - (-6) = 13.
Constant b = 13.
Formula: nth term = -3n + 13.
Check:
n=1: -3 + 13 = 10 ✅
n=2: -6 + 13 = 7 ✅
n=3: -9 + 13 = 4 ✅
Common Student Pitfalls and How to Avoid Them
Mixing up n and the term value: Remember, n is the position number (st, 2nd...), not the term itself.
Getting the sign wrong for decreasing sequences: If the sequence goes down, your common difference (a) is negative.
Forgetting to check: The check takes 10 seconds and can save you from losing easy marks. Always do it!
- Misreading the question: Sometimes you are asked for the 10th term. In this case, find the nth term formula first, then substitute
n=10.
Practice Questions (With Answers at the Bottom)
Try these to build your confidence. All sequences are linear.
Find the nth term: 4, 9, 14, 19, 24...
Find the nth term: 21, 17, 13, 9, 5...
Find the 20th term of the sequence: 6, 10, 14, 18, 22...
A sequence is defined by the nth term 5n - 1. What are the first three terms?
- Challenge (Edexcel/AQA Style): The nth term of a sequence is
8 - 2n. Is the number -42 in the sequence? Show your working.
Why This Topic Matters
Mastering the nth term for linear sequences builds the essential algebraic thinking you need for:
Quadratic sequences (at GCSE).
Graphs of linear functions (y = mx + c is just the nth term in disguise!).
- Real-world problem solving, like calculating costs over time or patterns in data.
Summary
The key is to be systematic:
Find a (the common difference).
Write an and compare it to the sequence to find b.
Write an + b.
- CHECK!
Stick to these steps, and you'll consistently earn marks on this frequent GCSE question.
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Answers to Practice Questions:
5n - 1 (Common difference 5, adjustment -1)
-4n + 25 (Common difference -4, adjustment +25)
- 82 (nth term is
4n + 2. For n=20: (4 x - 20) + 2 = 82)
4, 9, 14 (n=1: 4, n=2: 9, n=3: 14)
- Yes, it is the 25th term. Solve
8 - 2n = -42.-2n = -50, son = 25. Since n is a positive integer, -42 is in the sequence.