Geometric Sequence Calculator

About this calculator

In a geometric sequence, each term is the previous one multiplied by a constant ratio r. The nth term is aₙ = a₁·r^(n−1), and the sum of the first n terms is Sₙ = a₁·(1 − rⁿ)/(1 − r) for r ≠ 1 (with the special case Sₙ = a₁·n when r = 1). When |r| < 1, the sum of the infinite series converges to S = a₁/(1 − r) as n grows without bound.

Students use this to work through algebra and precalculus problems on sequences and series, while it also models real growth-and-decay processes: compound interest and investment growth, bacterial population doubling, radioactive decay (as a decreasing geometric sequence), and the branching structure behind algorithm complexity analysis in computer science. Anyone modeling a quantity that grows or shrinks by a fixed percentage each step — from a biologist tracking a culture to a programmer estimating recursive call counts — relies on this formula.