A-Level Maths: Sequences & Series Revision

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What is the difference between an arithmetic and a geometric sequence?

Sequences & Series is a core part of A-Level Maths. Revise the key concepts and common mistakes below, then lock them in with the free games.

Key concepts

Arithmetic sequenceConstant difference d between terms. nth term: uₙ = a + (n−1)d. Sum: Sₙ = (n/2)(2a + (n−1)d).

Geometric sequenceConstant ratio r between terms. nth term: uₙ = ar^(n−1). Sum: Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.

Sum to infinityFor |r| < 1, infinite geometric series converges: S∞ = a/(1 − r). Diverges if |r| ≥ 1.

Sigma notationΣ symbol denotes summation: Σₖ₌₁ⁿ aₖ means a₁ + a₂ + ... + aₙ.

Recurrence relationDefines each term using earlier terms: uₙ₊₁ = f(uₙ). E.g., uₙ₊₁ = 2uₙ + 1, u₁ = 3.

Convergence vs divergenceSequence converges if it tends to a finite limit; diverges otherwise. Series converges if partial sums approach a finite limit.

nth termu_n = a + (n-1)d for first term a.

Sum formulaS_n = n/2 [2a + (n-1)d].

Equivalent sumS_n = n/2 (a + l) where l is the last term.

Identifying dd = u_{n+1} - u_n; constant for arithmetic.

Σ notationΣ from k=1 to n means add up terms as k varies.

Binomial theorem(a+b)ⁿ = Σ C(n,r) a^(n-r) b^r for non-negative integer n.

nCr coefficientC(n,r) = n!/[r!(n-r)!] counts ways to choose r from n.

Pascal's triangleEach entry is sum of the two above; row n gives C(n,0)…C(n,n).

What your exam board covers

Where Sequences & Series sits in each exam board's specification. Tap a board to see what it lists.

AQASequences and seriesSpec 7357

Part of: Pure mathematics

Source: AQA A-level Mathematics (7357) specification — Subject content, accessed 23 June 2026.

CCEASequences and seriesSpec 2210

Part of: AS 1 — Pure mathematics

Source: CCEA A-Level Mathematics specification, accessed 23 June 2026.

EdexcelSequences and seriesSpec 9MA0

Part of: Pure mathematics

Source: Pearson Edexcel A level Mathematics (9MA0) specification, accessed 23 June 2026.

OCRSequences and seriesSpec H240

Part of: Pure mathematics

Source: OCR A Level Mathematics A (H240) specification, accessed 23 June 2026.

Learn Sequences & Series step by step

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Common mistakes to avoid

Questions where students often pick the tempting wrong answer — make sure you know the right one:

What is the difference between an arithmetic and a geometric sequence?✗ Arithmetic sequences involve only addition and geometric sequences only multiplication, so 1, 2, 4 is arithmetic. ✓ Arithmetic has a constant difference between terms; geometric has a constant ratio.

Does the sequence 1, 1/2, 1/3, 1/4, ... ever reach 0?✗ Eventually a term will be so small it equals 0. ✓ No — it approaches 0 as the limit but no term is ever exactly 0.

Practise Sequences & Series — free games

Test yourself with these quick revision games for this topic:

▶ Two Truths, One Lie ▶ Reveal Race ▶ Connections Grid ▶ Falling Words ▶ Word Web

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Sequences & Series

Key concepts

What your exam board covers

Common mistakes to avoid

Practise Sequences & Series — free games

More A-Level Maths topics