Overview
Sequences where each term follows a fixed rule — either adding (arithmetic) or multiplying (geometric).
Arithmetic progression (AP)
Common difference d added each time.
- nth term: a + (n − 1)d
- Sum of n terms: Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)
Geometric progression (GP)
Common ratio r multiplied each time.
- nth term: a rⁿ⁻¹
- Sum of n terms: Sₙ = a(rⁿ − 1) ÷ (r − 1)
- Sum to infinity (only if |r| < 1): S∞ = a ÷ (1 − r)
Worked example
GP with a = 2, r = 3: 5th term = 2 × 3⁴ = 162. AP with a = 5, d = 3: sum of first 10 terms = 10/2 [2(5) + 9(3)] = 5 × 37 = 185.
Common mistakes
- Using the sum-to-infinity formula when |r| ≥ 1 — it only works if |r| < 1.
Exam tips
- Spot the type first: constant difference = AP; constant ratio = GP.