A-Level Maths: Sequences and Series

Sequences and series questions at A-Level are engineered around a few precise formulas and one crucial convergence condition — and the marks go to candidates who can set up the right equations from partial information. This quiz builds exactly that skill.

The arithmetic questions use the exam's favourite construction: recovering the common difference from two non-adjacent terms (the 5th term 14 and the 12th term 35, giving 7d = 21), and finding a specific term from the standard nth-term formula. Sigma notation appears as a computation — evaluating the sum of (3k − 2) from 1 to 10 by recognising it as an arithmetic series and pairing first and last terms.

The geometric questions mirror the same reverse-engineering style: finding the first term from the 2nd and 4th terms (where r² = 9 requires the positive-terms condition to select r = 3), computing the sum to infinity of the alternating series 10 − 5 + 2.5 − ⋯ with its negative ratio, and identifying which series converge at all. The convergence condition itself gets a dedicated question — sharpened during verification into its cleanest form: |r| < 1 as the necessary and sufficient condition, with r ≠ 1 exposed as necessary but nowhere near sufficient.

The binomial expansion questions cover both the mechanics — the coefficient of x² in (1 − 3x)⁶, where the (−3)² must be squared along with the binomial coefficient — and the structural properties of (1 + x)ⁿ for positive integer n: term count and coefficient symmetry, against tempting false claims.

Every explanation shows the full setup and solution, and the distractors are built from the standard slips: off-by-one term counts, forgetting to square the coefficient inside the bracket, and dividing by the wrong denominator in the sum-to-infinity formula.

• Recover arithmetic sequence parameters from two given terms
• Evaluate sums written in sigma notation
• Solve geometric sequence problems, selecting the valid ratio from conditions
• Apply the sum to infinity with |r| < 1 as the exact convergence condition
• Extract binomial expansion coefficients including negative bracket terms

Topic scope follows section D (Sequences and series) of the DfE's prescribed AS and A level mathematics subject content — 100% common across all exam boards: the binomial expansion, arithmetic and geometric sequences and series, sum to infinity, and sigma notation.