Arithmetic vs geometric sequences

Progressions are number sequences with a fixed pattern between one term and the next. The two most studied are arithmetic (AP) and geometric (GP) — the difference is how you move from one term to the next.

AP — arithmetic progression

Each term adds the same ratio: 2, 5, 8, 11… (ratio 3).

• General term: aₙ = a₁ + (n−1)·r
• Sum of first n: Sₙ = n·(a₁ + aₙ) / 2

Example: the 10th term of 2, 5, 8… is 2 + 9×3 = 29. The sum of the first 10 is 10×(2+29)/2 = 155.

GP — geometric progression

Each term multiplies by the same ratio: 2, 6, 18, 54… (ratio 3).

• General term: aₙ = a₁·rn−1
• Sum of first n: Sₙ = a₁·(rⁿ − 1)/(r − 1)

Example: the 5th term of 2, 6, 18… is 2×3⁴ = 162.

Where each appears

The GP describes compound interest and anything growing by multiplication (exponential). The AP describes fixed installments and simple interest.

FAQ

How do I find the ratio? In an AP, subtract neighbors (5 − 2 = 3). In a GP, divide (6 ÷ 2 = 3).

Is there an infinite GP sum? Yes, when |r| < 1: S = a₁ / (1 − r).