Sequences and Series
Sequences and Series rewards precision. The exam tests arithmetic and geometric progressions, convergence, sigma notation, the standard summation results, and binomial expansion for both positive-integer and general rational indices. Most marks disappear through off-by-one counting rather than conceptual difficulty.
Arithmetic progressions
- nth term uₙ = a + (n − 1)d; sum Sₙ = n/2[2a + (n − 1)d] = n/2(a + l) where l is the last term.
- Three terms in AP are conveniently written a − d, a, a + d.
Geometric progressions
- uₙ = ar^(n−1); Sₙ = a(1 − rⁿ)/(1 − r), equivalently a(rⁿ − 1)/(r − 1).
- A sum to infinity exists only when |r| < 1, and then S∞ = a/(1 − r).
- Three terms in GP: a/r, a, ar.
- A common problem type, "smallest n such that the sum exceeds 1000", is solved with logarithms, not trial.
Sigma notation and standard results
Σ1 = n, Σr = n(n + 1)/2, Σr² = n(n + 1)(2n + 1)/6. Split sums term by term and factor constants out.
Binomial expansion
- Positive integer n: (a + b)ⁿ = Σ C(n, r) a^(n−r) b^r, giving n + 1 terms.
- Any rational n with |x| < 1: (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + …
- The general term of (a + bx)ⁿ is C(n, r) a^(n−r)(bx)^r, useful for "find the coefficient of x⁴" questions.
Worked example
The second term of a GP is 6 and the fifth is 3/4. Find r, a and S∞. Dividing, r³ = u₅/u₂ = (3/4)/6 = 1/8, so r = 1/2. Then a = u₂/r = 6/(1/2) = 12. Since |r| < 1 the sum converges: S∞ = 12/(1 − 1/2) = 24.
Traps
- Counting terms wrongly when the series starts at r = 0 or at a later index.
- Applying S∞ when |r| ≥ 1, especially with a large negative r.
- Quoting a general binomial expansion of (1 + x)ⁿ without stating its range of validity |x| < 1.