🔢 Geometric Sequence Calculator – nth Term, Sums & More
A geometric sequence is one of the most fundamental structures in mathematics. Each term is obtained by multiplying the previous term by a fixed constant called the common ratio (r). Whether you are studying for an exam, designing exponential growth models, or exploring financial compounding, this calculator covers every major geometric sequence operation in one place.
📐 Core Formulas at a Glance
Nth Term
aₙ = a₁ × r^(n−1)Finds the value of any specific term directly without computing all previous terms.
Finite Sum (Sₙ)
Sₙ = a₁ × (1 − rⁿ) / (1 − r)Sum of the first n terms. When r = 1, use Sₙ = n × a₁ instead.
Infinite Sum (S∞)
S∞ = a₁ / (1 − r) when |r| < 1Only valid when the absolute value of the common ratio is less than 1.
Solve for Ratio
r = (aⱼ / aᵢ)^(1 / (j − i))Derives the common ratio and first term when two non-consecutive terms are known.
🧮 Calculation Modes Explained
Nth Term
Use this mode when you know a₁ (first term) and r (common ratio) and want to find the value at a particular position. The explicit formula aₙ = a₁ × r^(n−1) gives you the result without computing every intermediate term. For example, with a₁ = 3 and r = 2, term 6 is 3 × 2⁵ = 96.
Generate Sequence Terms
Lists the first n terms of your sequence. This is ideal for pattern recognition and classroom exercises. You can generate up to 1,000 terms, with the table showing up to 20 at a time for readability. Negative ratios produce alternating-sign sequences (e.g., 3, −6, 12, −24, …), while ratios between −1 and 1 produce decaying sequences.
Finite Geometric Sum
Computes the sum of the first n terms using Sₙ = a₁ × (1 − rⁿ) / (1 − r). The special case where r = 1 reduces to Sₙ = n × a₁ (an arithmetic, not geometric, sum), and this calculator handles it automatically.
Infinite Sum and Convergence
An infinite geometric series converges only when |r| < 1. In that case, S∞ = a₁ / (1 − r). If |r| ≥ 1, the series diverges — its partial sums grow without bound (or oscillate indefinitely), and no finite infinite sum exists. The calculator displays a clear convergence badge so you always know which case applies.
Solve for Common Ratio
When you know the values at two different positions in the sequence (but not the first term or ratio), this mode recovers the common ratio via r = (aⱼ / aᵢ)^(1 / (j − i)), then back-calculates the first term. This is useful for curve-fitting, financial modeling, and reverse-engineering geometric patterns in data.
💡 Real-World Applications
Geometric sequences appear throughout science, finance, and engineering:
- Compound interest — a bank balance growing at a fixed interest rate follows a geometric sequence where the ratio equals
1 + interest rate. - Population growth and decay — exponential models are discrete geometric sequences.
- Radioactive decay — each half-life multiplies the remaining quantity by 0.5.
- Signal attenuation — each stage of a chain reduces power by a fixed fraction.
- Fractals and self-similar structures — geometric ratios define self-similar scaling.
⚠️ Accuracy and Limitations
All calculations use standard 64-bit floating-point arithmetic. For very large exponents (e.g., r = 10, n = 400), results may overflow to Infinity; for very small ratios and large n, they may underflow to 0. The precision control (0–10 decimal places) lets you fine-tune rounding. For exact fractional results, consider working with fractions and verify with the decimal output. The "Solve Ratio" mode requires both known values to have a consistent real ratio — if no real solution exists (e.g., even-index difference with sign change), the calculator reports this clearly.