About the Geometric Progression Calculator

A geometric progression multiplies each term by a constant ratio r: a, ar, ar², ar³, …. The nth term is a·r^(n−1), and the sum of the first n terms is a(1−rⁿ)/(1−r) — with the special case r = 1 giving simply a·n. Growth by a fixed percentage is always geometric: a 5% annual return on $1,000 produces the sequence 1,050, 1,102.50, 1,157.63, …

Convergence and divergence

When the ratio's magnitude is below 1 — like r = 1/2 — each term shrinks and the infinite sum converges to a/(1−r). The classic example: 1 + 1/2 + 1/4 + 1/8 + … = 2. With |r| ≥ 1 the series grows without bound, which is why doubling sequences (the chessboard wheat problem) explode so quickly. The calculator reports both the finite sum and, when it exists, the infinite limit.