Geometric Sequence

The geometric sequence calculator finds the n-th term, the common ratio and the sum of the first n terms, as well as the sum of an infinite geometric series. A sequence can be given by its first term and common ratio, by the first and the n-th term, or by any two terms. When two ratios fit (q and −q), the calculator solves both cases. The answer is exact — in fractions, with a step-by-step solution and a table of the first terms.

What is a geometric sequence

A geometric sequence (geometric progression) is a sequence of non-zero numbers in which each term after the first is obtained by multiplying the previous term by the same number q. This number is called the common ratio.

b_n+1 = b_n · q,   b₁ ≠ 0,   q ≠ 0

For example, 3, 6, 12, 24, 48, … is a geometric sequence with b₁ = 3 and q = 2. When q < 0, the signs of the terms alternate: 1, −2, 4, −8, …

Main formulas

The absolute value of every term except the first is the geometric mean of its neighbors — this is where the name comes from.

Example 1. b₁ = 3, q = 2. Find b₁₀ and S₁₀.
b₁₀ = 3 · 2⁹ = 1536
S₁₀ = 3 · (2¹⁰ − 1) / (2 − 1) = 3069

How to find the ratio from two terms

The ratio of two terms equals q to the power of the difference of their positions: b_m / b_k = q^m−k. Then take the root of degree m − k. If that degree is even and the ratio is positive, two common ratios fit: q and −q.

Example 2. b₁ = 2, b₅ = 162.
q⁴ = 162 / 2 = 81 ⇒ q = 3 or q = −3.
For q = 3: S₅ = 242, for q = −3: S₅ = 122.

Infinite geometric series

If |q| < 1, the terms approach zero and the partial sums S_n approach a finite number S = b₁ / (1 − q). If |q| ≥ 1, the series has no finite sum.

Example 3. 8 + 4 + 2 + 1 + … : b₁ = 8, q = 1/2.
S = 8 / (1 − 1/2) = 16

This formula turns a repeating decimal into a fraction: 0.(3) = 0.3 + 0.03 + … = 0.3 / (1 − 0.1) = 1/3.