Geometric Sequence Calculator
Find any term, the partial sum, and the infinite sum of a geometric sequence. Enter the first term, common ratio, and the desired term number to see the results and a preview of the sequence.
A geometric sequence (or geometric progression) is a sequence where each term is found by multiplying the previous term by a fixed constant called the common ratio (r). For example, the sequence 3, 6, 12, 24, 48 has a common ratio of 2.
The nth term is aₙ = a₁ × r^(n-1). The sum of the first n terms is Sₙ = a₁(1 - r^n) / (1 - r) when r is not equal to 1. If |r| < 1, the infinite geometric series converges to S∞ = a₁ / (1 - r). For example, the series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2.
Geometric sequences model exponential growth and decay, including compound interest, population growth, radioactive decay, and the physics of bouncing balls. They are also fundamental to the study of infinite series, fractals, and Zeno's paradox.