Sheet № 87 · Foundation + Higher · AQA · Edexcel · OCR
Nth Term of Linear Sequences –
Finding the nth term of a linear sequence is a key algebra skill at GCSE. It lets you write a rule that generates any term in a sequence without having to list them all, and it is tested on both Foundation and Higher papers across all exam boards.
§Key definitions
Question:
Find the nth term of the sequence 4, 9, 14, 19, 24, ...
Answer:
Yes, 52 is in the sequence (it is the 15th term).
Q1 (Foundation):
Find the nth term of the sequence 7, 11, 15, 19, 23, ...
Q2 (Foundation):
Find the 20th term of the sequence with nth term 6n - 5.
Q3 (Higher):
The nth term of a sequence is 8n - 3. Prove that 150 is not a term in this sequence.
§Formulas to memorise
nth term = dn + (a - d), where d = common difference and a = first term
Equivalent form: nth term = a + (n - 1)d
Compare the first term of the sequence with d times 1 to find the adjustment c = a - d.
Check by substituting n = 1, 2, 3 to verify you get the original terms.
Worked example
Find the nth term of the sequence 4, 9, 14, 19, 24, ...
Working:
⚠ Common mistakes
- ✗Using the wrong sign for the common difference. If the sequence decreases, d is negative. Always subtract in the correct order: next term minus current term.
- ✗Forgetting to find the adjustment c. Writing the answer as just "5n" when the nth term is 5n - 1 loses a mark. You must include the constant.
- ✗Confusing the first term with the zeroth term. The adjustment is found by comparing d times 1 with the first term, not d times 0.
✦ Exam tips
- →To check if a number belongs to the sequence, set the nth term formula equal to that number and solve for n. If n is a positive integer, it is in the sequence.
- →The common difference d is always the coefficient of n in the nth term formula. Use this as a quick sanity check.
- →For sequences with negative common differences, be extra careful with signs when calculating the adjustment.