Sequences and Series
Sequences and Series
Sequences and series questions test arithmetic and geometric progressions, sums to n terms and to infinity, sigma notation, and the binomial expansion of (1 + x)ⁿ. The exam expects clean algebraic handling of a, d, r and n, often through simultaneous equations formed from two given terms.
Arithmetic progressions
nth term uₙ = a + (n − 1)d. Sum Sₙ = (n/2)(2a + (n − 1)d) = (n/2)(first + last). If two terms are given, subtract their equations to eliminate a and find d; never substitute before simplifying.
Geometric progressions
nth term uₙ = ar^(n−1). Sum Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. The sum to infinity exists only when |r| < 1, and then S∞ = a/(1 − r). For a converging GP, S∞ − Sₙ = arⁿ/(1 − r), which is the tail you discard.
Binomial expansion
For positive integer n, (a + b)ⁿ = Σ ⁿCᵣ a^(n−r) b^r, with general term ⁿCᵣ a^(n−r)b^r. Use the general term to pick out one coefficient rather than expanding everything. For |x| < 1 and any rational n, (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + …, valid only inside the stated range.
Worked example
A geometric progression has first term 24 and sum to infinity 32. Find r and the sum of the first four terms. S∞ = a/(1 − r) gives 32 = 24/(1 − r), so 1 − r = 24/32 = 3/4 and r = 1/4, which satisfies |r| < 1 as required. Then S₄ = 24(1 − (1/4)⁴)/(1 − 1/4) = 32(1 − 1/256) = 32 × 255/256 = 255/8 = 31.875. The answer is sensibly just below S∞ = 32, which confirms the arithmetic.
Traps that cost marks
- Using r = 1/4 but forgetting to check |r| < 1 before quoting a sum to infinity.
- Mixing up n − 1 and n in the nth term of a GP.
- Sign slips in Sₙ = a(1 − rⁿ)/(1 − r) when r is negative.
- Expanding a binomial fully when the question only asked for one term.
- Quoting a binomial expansion for (1 + x)ⁿ without giving the range of validity.