GCSE Maths: Sequences and the nth Term

Sequences questions reward pattern recognition backed by algebra, and they appear on GCSE papers at every grade from 2 to 9. This quiz works through the full range the specification names, from term-to-term rules to the Higher tier quadratic nth term.

The opening questions cover the descriptive layer: stating the term-to-term rule of an arithmetic sequence and recognising the special number families — square numbers, cube numbers and triangular numbers — that examiners expect you to know on sight, including the triangular numbers' formula n(n+1)/2. From there the quiz moves to the core skill of finding the nth term of a linear sequence from its common difference and first term, and the reverse skill of evaluating a given nth-term expression such as 3n² + 2 at a specific position.

Geometric sequences get their own questions — identifying a constant ratio between consecutive terms and distinguishing geometric from arithmetic behaviour — as does the decreasing linear sequence 10, 7, 4, 1, whose negative common difference produces the nth term 13 − 3n, a form students frequently get backwards.

The Higher tier material centres on quadratic sequences: computing first and second differences, using the fact that a constant second difference signals a quadratic rule, and applying the relationship that the second difference equals 2a to recover the leading coefficient. These two facts together are the fast route to any quadratic nth term, and the explanations spell out exactly how they combine.

Every question is self-contained with the sequence written out in full, and the distractors encode the usual slips: off-by-one position errors, swapped difference and first term, and halving instead of doubling when recovering a from the second difference.

• State term-to-term and position-to-term rules for arithmetic sequences
• Recognise square, cube and triangular numbers and the formula n(n+1)/2
• Find the nth term of a linear sequence and evaluate an nth-term expression at a given position
• Identify geometric sequences by their constant ratio
• Use first and second differences to analyse quadratic sequences, applying second difference = 2a (Higher)

Topic scope follows the Algebra strand of the Department for Education's GCSE mathematics subject content: generating terms of a sequence from term-to-term or position-to-term rules, recognising arithmetic, geometric and special-number sequences, and deducing expressions for the nth term of linear and (Higher) quadratic sequences.