Geometric Sequence Calculator

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant called the common ratio (r). Example: 2, 6, 18, 54 with r = 3.

Use aₙ = a₁ × r^(n−1). For example, the 5th term of a sequence with a₁=2 and r=3 is 2 × 3⁴ = 162.

If |r| < 1, the terms approach zero and the infinite sum converges to a₁/(1−r). For example, 1 + 1/2 + 1/4 + 1/8 + … = 1/(1−0.5) = 2.

When r > 1, the terms grow without bound. The sequence is divergent and the infinite sum does not exist. The partial sum Sₙ still grows as n increases.

A geometric sequence is just the list of terms: a₁, a₁r, a₁r², … A geometric series is the sum of those terms: a₁ + a₁r + a₁r² + …