Sheet № 17 · Foundation + Higher · AQA · Edexcel · OCR
Sequences and Nth Term –
Sequences are patterns of numbers that follow a rule, and finding the nth term is one of the most satisfying skills in GCSE Maths. This topic appears on both Foundation and Higher tier papers and is tested by AQA, Edexcel, and OCR every year. You might be asked to continue a sequence, find the nth term rule for a linear (arithmetic) seque
§Key definitions
Question:
Find the nth term of the sequence 2, 9, 16, 23, 30, ...
Check:
n = 4 → 7(4) − 5 = 23 ✓
Answer:
nth term = 7n − 5
Follow-up:
Is 100 in this sequence?
Q1 (Foundation):
Find the nth term of the sequence 6, 10, 14, 18, 22, ...
§Formulas to memorise
nth term = dn + (a − d), where d is the common difference and a is the first term
nth term = a + (n − 1)d
3, 7, 11, 15, 19, ... (common difference = 4)
20, 17, 14, 11, 8, ... (common difference = −3)
Work out the common difference (d) — by subtracting consecutive terms. For 5, 8, 11, 14: d = 8 − 5 = 3.
Write the "times table" part. — The nth term starts with dn, so here it starts with 3n.
Compare 3n with the sequence. — When n = 1, 3n = 3, but the first term is 5. The difference is 5 − 3 = 2.
Add the adjustment. — The nth term = 3n + 2.
Check with another term. — When n = 3, 3(3) + 2 = 11 ✓.
To find the 50th term: — substitute n = 50 into the formula.
Worked example
Find the nth term of the sequence 2, 9, 16, 23, 30, ...
Working:
⚠ Common mistakes
- ✗Getting the sign of the adjustment wrong. If d = 5 and the first term is 3, then 5n gives 5 when n = 1, so the adjustment is 3 − 5 = −2, not +2. Always subtract the value of dn at n = 1 from the actual first term.
- ✗Confusing the common difference with the first term. The common difference is found by subtraction between consecutive terms, not by looking at the first number.
- ✗Forgetting that n must be a positive integer. When checking membership, a decimal or negative value of n means the number is not in the sequence.
- ✗Assuming all sequences are arithmetic. If the differences between terms are not constant, the sequence is not linear — it might be quadratic or geometric. Check before applying the linear formula.
- ✗Miscounting term positions. The first term corresponds to n = 1, not n = 0.
✦ Exam tips
- →Always show how you found d. Write the subtraction explicitly, e.g., "d = 9 − 5 = 4." This earns the first method mark.
- →Check your formula against at least two terms from the original sequence. If both match, your formula is correct.
- →"Is x in the sequence?" questions are very common. Set the formula equal to x, solve for n, and state clearly whether n is a whole number.
- →On Higher papers, you may meet decreasing sequences where d is negative. The method is identical; just be careful with signs.