Geometric Progression

What is a Geometric Progression?

In a GP, you reach each term by multiplying the previous one by the ratio q. That constant multiplication is what defines the sequence.

• General term: aₙ = a₁ × q^(n−1)
• Sum (q ≠ 1): Sₙ = a₁ × (q^n − 1) ÷ (q − 1)
• Sum (q = 1): Sₙ = n × a₁

Geometric progression: formula and example

A geometric progression (GP) is a sequence where each term is the previous one multiplied by a fixed ratio q. The general term is aₙ = a₁·q^(n-1). The sum of the first n terms (for q ≠ 1) is Sₙ = a₁·(qⁿ - 1)/(q - 1). With a₁ = 2, q = 3, n = 5: a₅ = 2·3⁴ = 162 and S₅ = 2·(243 - 1)/2 = 242.

When |q| < 1, the infinite sum converges: S∞ = a₁/(1 - q). Example: 1 + 1/2 + 1/4 + ... = 2 — this is how Zeno's paradox of Achilles and the tortoise is resolved: infinitely many shrinking intervals add to a finite distance.

Applications

GPs model compound interest (the balance after each period is a GP of ratio 1+i), exponential depreciation, radioactive decay (half-life: ratio 1/2 per half-life), Malthusian population growth, and fractals like the Cantor set and Koch snowflake — whose perimeter is infinite while the enclosed area is finite, precisely because perimeter is a divergent GP (q > 1) and area is a convergent one. In music, octaves form a GP of ratio 2 and the 12-tone equal temperament uses ratio q = 2^(1/12).

FAQ

GP vs AP? In an arithmetic progression each term adds a constant difference; in a GP it multiplies by a constant ratio. APs grow linearly, GPs grow (or shrink) exponentially.

Can the ratio be negative? Yes — terms then alternate sign (e.g., 1, -2, 4, -8, ...). The infinite sum still converges if |q| < 1.

What if q = 1? All terms are equal to a₁ and Sₙ = n·a₁. The formula Sₙ = a₁·(qⁿ - 1)/(q - 1) is undefined (0/0), so use the simple multiplication.

Why does S∞ exist only when |q| < 1? Because qⁿ → 0 as n → ∞ only when |q| < 1. If |q| ≥ 1, terms do not shrink fast enough and the sum diverges.