Sequences
Sequences
Distinguish arithmetic sequences (constant difference) from geometric sequences (constant ratio), and know the general term, the sum formulas, and the behavior of each. For arithmetic, aₙ=a₁+(n−1)d and Sₙ=n(a₁+aₙ)/2. For geometric, aₙ=a₁q^{n−1}, with Sₙ=a₁(1−qⁿ)/(1−q) for q≠1 and Sₙ=na₁ for q=1. Many problems use the relation aₙ=Sₙ−Sₙ₋₁ for n≥2, together with the separately checked value a₁=S₁.
Traps: applying Sₙ−Sₙ₋₁ at n=1; forgetting that a geometric ratio may be negative or zero; using the sum formula when q=1; and misreading “the sum of the first n terms” as an infinite sum. Recurrence questions often require writing the first few terms to detect a pattern, then proving it by induction or by subtracting consecutive relations. Monotonicity is best checked by comparing aₙ₊₁−aₙ with 0, not by guessing from a graph.
Worked example: if Sₙ=n², then for n≥2, aₙ=Sₙ−Sₙ₋₁=n²−(n−1)²=2n−1. Also a₁=S₁=1, which fits the same formula, so aₙ=2n−1 for all n≥1. Rule of thumb: whenever a sum Sₙ is given, compute a₁ separately, derive aₙ for n≥2, and then decide whether the formula extends to n=1.