Sequences and series
Sequences and series examines arithmetic and geometric progressions, sigma notation, sums to infinity and the binomial expansion. You should be able to derive and apply the standard formulae, decide when an infinite sum converges, and extract a specified coefficient from a binomial expansion without expanding everything.
Arithmetic and geometric progressions
For an arithmetic progression with first term a and common difference d: the nth term is a + (n − 1)d and the sum of n terms is (n/2)(2a + (n − 1)d). For a geometric progression with common ratio r: the nth term is ar^(n−1), the sum of n terms is a(1 − rⁿ)/(1 − r), and the sum to infinity is a/(1 − r), valid only when |r| < 1.
Sigma notation and the binomial theorem
The symbol Σ denotes a sum whose index runs from a lower to an upper limit. The binomial expansion of (a + b)ⁿ for a positive integer n is the sum over r from 0 to n of C(n, r) a^(n−r) b^r, where C(n, r) = n!/(r!(n − r)!). The general term is C(n, r) a^(n−r) b^r, so a single coefficient can be found directly.
Worked example
Find the coefficient of x³ in the expansion of (2 + 3x)⁷.
The general term is C(7, r)·2^(7−r)·(3x)^r. For x³ take r = 3: C(7, 3)·2⁴·3³ = 35 × 16 × 27 = 15 120. Therefore the coefficient of x³ is 15 120.
Traps that cost marks
- Using the sum-to-infinity formula when |r| ≥ 1, where the series diverges.
- Miscounting terms: a series from r = 1 to n has n terms, but from r = 0 to n it has n + 1.
- Forgetting to raise the numerical part of the bracket to the appropriate power.
- Dropping a negative sign when the common ratio is negative and the power is odd.
- Confusing the nth term with the sum of n terms in a worded modelling question.