Geometric Progression Solver — Find nth Term, Sum & Ratio [2026]

A geometric progression (GP) is a sequence where each term is found by multiplying the previous term by a fixed number called the common ratio. For example: 2, 6, 18, 54 has ratio = 3. Each term is 3 times the previous one.

It uses three standard GP formulas: nth term = a × r^(n-1), Sum of n terms = a × (r^n - 1) / (r - 1) when r ≠ 1, and Sum to infinity = a / (1 - r) when |r| < 1. All formulas are displayed alongside results.

The sum to infinity exists only when the absolute value of the common ratio is less than 1 (|r| < 1). This means the terms get progressively smaller and the sum converges to a finite value. For example, 1 + 0.5 + 0.25 + 0.125 + ... converges to 2.

Yes. A negative common ratio creates an alternating sequence where terms switch between positive and negative. For example, a=1, r=-2 gives 1, -2, 4, -8, 16, -32, and so on.

Yes. The common ratio can be any non-zero number including decimals and fractions. For example, r=0.5 creates a halving sequence (10, 5, 2.5, 1.25, ...) and r=1.05 models 5% compound growth.

The solver can calculate and display sequences of any reasonable length. For very large values of n or r, the numbers grow extremely fast (exponential growth), so the display focuses on the most useful information.