Geometric Sequence Calculator

In a geometric sequence each term is the previous one multiplied by the same number, the common ratio r: 3, 6, 12, 24… Geometric sequences describe compound growth, halving and repeated percentages. This calculator finds any term, the sum of the first n terms and, when |r| < 1, the sum of the whole infinite series. It can also recover the first term and ratio from two known terms.

How to use this tool

1. Choose whether you know the first term and ratio, or two terms and their positions.
2. Enter the values and the term number n.
3. Read aₙ, Sₙ, the infinite sum and the formulas.
4. Use the table of the first 20 terms to see how fast the sequence grows or shrinks.

Formula

aₙ = a₁ · r^(n − 1) · Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for r ≠ 1, n·a₁ for r = 1 · S∞ = a₁ ÷ (1 − r) when |r| < 1

From two terms, r^(k − m) = aₖ ÷ aₘ. With an odd number of steps there is one real ratio; with an even number, both r and −r fit and a negative quotient has no real solution. The infinite sum converges only when the terms shrink towards zero, which needs |r| < 1.

Example

For 3, 6, 12, … (a₁ = 3, r = 2), the 10th term is 3 × 2⁹ = 1,536 and the first 10 terms add to 3 × (2¹⁰ − 1) = 3,069. The series 1 + ½ + ¼ + … has a₁ = 1 and r = 0.5, so its infinite sum is 1 ÷ (1 − 0.5) = 2.