Sequences, Series and the Binomial Theorem
Core ideas
Arithmetic sequences have a constant difference d; the nth term is a + (n−1)d and the sum of n terms is n/2[2a + (n−1)d]. Geometric sequences have a constant ratio r; the nth term is ar^(n−1) and the sum is a(1−rⁿ)/(1−r) for r ≠ 1. For |r| < 1 the sum to infinity is a/(1−r). The binomial theorem expands (a + b)ⁿ using ⁿCᵣ a^(n−r) b^r, with the general term ⁿCᵣ a^(n−r) b^r. Candidates should also recognise convergence conditions and use the binomial expansion for small approximations.
Traps
- Using n instead of n−1 in the nth-term formula.
- Applying the sum-to-infinity formula when |r| ≥ 1.
- Miscounting the number of terms between two given terms.
- Forgetting that the binomial coefficient counts from r = 0.
- Sign errors when b is negative, since b^r changes sign for odd r.
Worked example
Find the term in x³ in the expansion of (2 + 3x)⁵. The general term is ⁵Cᵣ · 2^(5−r) · (3x)^r. For x³, r = 3: ⁵C₃ · 2² · 3³ = 10 · 4 · 27 = 1080. So the coefficient is 1080.
Rule of thumb
Write the general term first every time; substituting the required power of x identifies r immediately and prevents coefficient slips.