A sequence is an ordered list of numbers that follow a rule. At KS3 you need to recognise patterns, continue sequences, and find the nth term — a formula that generates any term in the sequence directly without working through all the preceding terms.
What is a sequence?
A sequence is a list of numbers arranged in a definite order. Each number in the sequence is called a term. Terms are usually labelled 1st, 2nd, 3rd, … or written as T(1), T(2), T(3), …
Two key rules describe sequences:
- Term-to-term rule: tells you how to get from one term to the next (e.g. "add 4 each time").
- Position-to-term rule (nth term): tells you how to calculate any term directly from its position number n.
Types of sequences at KS3
| Type | Description | Example |
|---|---|---|
| Linear (arithmetic) | Constant difference between terms | 3, 7, 11, 15, … (add 4) |
| Geometric | Constant multiplier (ratio) between terms | 2, 6, 18, 54, … (multiply by 3) |
| Quadratic | Second difference is constant | 1, 4, 9, 16, 25, … (square numbers) |
| Fibonacci-type | Each term is the sum of the two before | 1, 1, 2, 3, 5, 8, 13, … |
At KS3 the main focus is linear sequences. Finding the nth term of quadratic sequences is introduced at GCSE.
Continuing a sequence using the term-to-term rule
Worked example 1
Continue the sequence: 5, 9, 13, 17, ____, ____
Find the difference between consecutive terms: 9 − 5 = 4, 13 − 9 = 4, 17 − 13 = 4.
Common difference = 4. Add 4 each time.
Next terms: 17 + 4 = 21, 21 + 4 = 25
Answer: 5, 9, 13, 17, 21, 25
Worked example 2: descending sequence
Continue the sequence: 30, 25, 20, 15, ____, ____
Common difference: 25 − 30 = −5 (subtract 5 each time).
Next terms: 15 − 5 = 10, 10 − 5 = 5
Answer: 30, 25, 20, 15, 10, 5
How to find the nth term of a linear sequence
The nth term of a linear sequence has the general form:
nth term = dn + c
where:
- d is the common difference (how much the sequence increases or decreases each step)
- c is a constant that adjusts the formula to match the actual sequence
Step 1 — Find the common difference d
Subtract any term from the term that follows it. This is d.
Step 2 — Find the constant c
Substitute n = 1 and T(1) (the first term) into dn + c = T(1) and solve for c.
Alternatively: write the sequence the formula d, 2d, 3d, 4d, … alongside the actual sequence; the difference at each position is the constant c.
Step 3 — Write the nth term as dn + c
Step 4 — Check by substituting n = 1, 2, and 3
Worked example 3
Find the nth term of: 7, 11, 15, 19, …
Step 1 — Common difference: 11 − 7 = 4. So d = 4.
Step 2 — Find c: Using 4n + c = T(n). When n = 1: 4(1) + c = 7, so c = 3.
Step 3 — nth term: 4n + 3
Step 4 — Check:
- T(1) = 4(1) + 3 = 7 ✓
- T(2) = 4(2) + 3 = 11 ✓
- T(3) = 4(3) + 3 = 15 ✓
Answer: nth term = 4n + 3
Worked example 4
Find the nth term of: 2, 5, 8, 11, …
d = 3 (add 3 each time).
When n = 1: 3(1) + c = 2, so c = −1.
nth term = 3n − 1
Check: T(1) = 2 ✓, T(4) = 3(4) − 1 = 11 ✓
Answer: nth term = 3n − 1
Worked example 5: decreasing sequence
Find the nth term of: 20, 17, 14, 11, …
d = −3 (subtract 3 each time).
When n = 1: −3(1) + c = 20, so c = 23.
nth term = −3n + 23 (often written as 23 − 3n)
Check: T(1) = 23 − 3 = 20 ✓, T(2) = 23 − 6 = 17 ✓
Answer: nth term = 23 − 3n
Using the nth term formula
Once you have the nth term, you can:
- Find any specific term by substituting the position number.
- Determine whether a given value is in the sequence by setting the nth term equal to that value and solving for n. If n is a positive integer, the value is in the sequence.
Worked example 6: find the 50th term
Using the sequence from Example 3 (nth term = 4n + 3):
T(50) = 4(50) + 3 = 200 + 3 = 203
Answer: the 50th term is 203
Worked example 7: is 47 in the sequence 4n + 3?
Set 4n + 3 = 47:
4n = 44
n = 11
11 is a positive integer, so yes — 47 is the 11th term.
Worked example 8: is 50 in the sequence 4n + 3?
Set 4n + 3 = 50:
4n = 47
n = 11.75
11.75 is not a whole number, so 50 is not in the sequence.
Summary of the method
| Step | Action | Example (sequence 7, 11, 15, 19, …) |
|---|---|---|
| 1 | Find d | 11 − 7 = 4 |
| 2 | Find c | 4(1) + c = 7 → c = 3 |
| 3 | Write nth term | 4n + 3 |
| 4 | Check with T(1), T(2), T(3) | 7, 11, 15 ✓ |
| Use it | Find T(n) or test a value | T(50) = 203; 47 is the 11th term |
Common mistakes to avoid
Mistake 1 — Using d as the first term rather than the common difference.
The coefficient of n is always the common difference, not T(1).
Mistake 2 — Not checking the formula.
Always substitute n = 1, 2, and 3 back into your formula to verify it produces the correct terms. Errors in finding c are caught this way.
Mistake 3 — Treating a non-integer n as valid.
If solving for n gives a decimal or negative number, the tested value is not in the sequence.
Mistake 4 — Forgetting the negative sign for decreasing sequences.
If the sequence goes down, d is negative. The nth term will be −dn + c, and it is easy to drop the negative sign.
How sequences fit the KS3 national curriculum
The Department for Education's KS3 mathematics programme of study requires pupils to "generate terms of a sequence from either a term-to-term or a position-to-term rule" and to "recognise arithmetic sequences and find the nth term." BBC Bitesize's KS3 algebra section positions the nth term as a key algebraic skill that bridges pattern recognition in Year 7 with quadratic sequences and geometric progressions at GCSE.
Frequently asked questions
What is the difference between a term-to-term rule and an nth term?
A term-to-term rule tells you how to get the next term from the current one (e.g. "add 5"). It is useful for continuing a sequence but requires you to calculate every term up to the one you want. The nth term is a formula that calculates any term directly from its position number — you can find the 100th term without listing all 99 previous terms.
Why does the nth term of a linear sequence always have n multiplied by the common difference?
In a linear sequence, the terms increase by d for each step. The 1st term is T(1), the 2nd is T(1) + d, the 3rd is T(1) + 2d, and so on. The nth term is T(1) + (n − 1)d = dn + (T(1) − d). Expanding this gives the dn + c form, where d is the common difference.
Can the nth term be a fraction or negative number?
The nth term formula can involve fractions and can produce negative values. For example, if the sequence is 1, 0.5, 0, −0.5, −1, …, the nth term is −0.5n + 1.5. However, the position number n must always be a positive integer (1, 2, 3, …) — fractions of a position have no meaning.
How do I know if a sequence is linear?
Calculate the differences between consecutive terms. If all the differences are the same (constant first difference), the sequence is linear. If the differences are not constant but the differences of the differences are (the second difference is constant), the sequence is quadratic.
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