geometric series

Also called: geometric sum, geometric progression

A sum in which each term is the previous one times the same factor q: a + aq + aq² + … For |q| < 1 the endless sum is a ÷ (1 − q); the first n terms add up to a(1 − qⁿ) ÷ (1 − q).

A geometric series adds terms that shrink (or grow) by the same factor q each step:

$a + aq + aq^2 + \dots = \frac{a}{1 - q} \quad (|q| < 1) \qquad a + aq + \dots + aq^{n-1} = a \cdot \frac{1 - q^n}{1 - q}$

It sits behind several product and finance formulas. Referrals: each new user brings k more, so a seed grows to seed ÷ (1 − k). Lifetime: with monthly churn c, a customer stays on average 1 + (1 − c) + (1 − c)² + … = 1 ÷ c months, so LTV ≈ margin ÷ churn. Discounting: payments worth 0.8, 0.64, 0.512… of their face value add up to an annuity factor.

The finite version gives capped values (LTV within 12 months), and with q = 1 + r it gives a savings pot, P × ((1 + r)ⁿ − 1) ÷ r.

Example

A Club member leaves 200 UAH a month; monthly churn is 5%, so q = 0.95.

Endless: 200 ÷ 0.05 = 4,000 UAH (20 months on average).
Within 12 months: 200 × (1 − 0.95¹²) ÷ 0.05 ≈ 1,839 UAH.
Referrals with k = 0.3: 100 families from ads become 100 ÷ 0.7 ≈ 142.9.

Common mistakes

Using a ÷ (1 − q) with q ≥ 1. The sum has no limit; the formula gives nonsense.
Trusting the endless sum. LTV = margin ÷ churn assumes churn never changes and ignores discounting.
Off-by-one terms. Check whether the first term is a or aq (first month counted or not).

Learn it in the course

Geometric series · Sums that shrink by the same factor: seed ÷ (1 − k) for referrals, LTV ≈ margin ÷ churn and lifetime 1 ÷ churn, the finite sum for annuities, capped LTV and a savings pot.