Sequences: arithmetic and geometric progressions
Two formula families
An arithmetic progression has constant difference d: aₙ = a₁ + (n − 1)d and Sₙ = n(a₁ + aₙ)/2 = n(2a₁ + (n − 1)d)/2. A geometric progression has constant ratio q: bₙ = b₁q^(n−1), Sₙ = b₁(qⁿ − 1)/(q − 1) for q ≠ 1, and the infinite sum b₁/(1 − q) exists only when |q| < 1.
Useful consequences: in an AP each term is the average of its neighbours; the arithmetic mean of a and b is (a + b)/2, while the geometric mean is √(ab), defined for ab ≥ 0. Three terms in AP are conveniently written a − d, a, a + d; three in GP as a/q, a, aq.
Traps. Mixing up n − 1 and n: the n-th term is n − 1 steps from the first. Using the infinite-sum formula when |q| ≥ 1. Forgetting that q may be negative (alternating signs) or that q = 1 needs separate handling. Confusing "sum of the first n terms" with "the n-th term". Reading an AP property into a GP.
Worked example. In an AP a₅ = 17 and a₁₁ = 41. Then 6d = 24, so d = 4 and a₁ = 17 − 4·4 = 1. Hence S₂₀ = 20(2·1 + 19·4)/2 = 10·78 = 780.
Rule of thumb: given two terms, subtract to obtain d (or divide to obtain q), then work backwards to a₁ or b₁.