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).
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.
How do you use a conversion graph between miles and kilometres?✗ Look only at the gradient of the line and apply it as a formula. ✓ Find the known value on its axis, draw a line to the graph, then read across (or down) to the other axis.
What is the relationship between the gradients of two perpendicular lines?✗ They are equal because perpendicular lines are 'similar' in direction. ✓ Their product is -1 — each is the negative reciprocal of the other.
Fully factorise 6x + 9.✗ 3(2x + 9), by only dividing the first term by the common factor. ✓ 3(2x + 3) — take out the common factor 3.
Practise Sequences & Series — free games
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