Geometric Sequence Calculator
Enter the first term (a₁), common ratio (r), and term number (n) to find the nth term, sum of first n terms, and—if |r| < 1—the infinite series sum.
What is a Geometric Sequence?
A geometric sequence (also called a geometric progression) is an ordered list of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). The general form is: a, ar, ar², ar³, …, arⁿ⁻¹. If r > 1, the sequence grows without bound (exponential growth); if 0 < r < 1, the terms shrink toward zero (exponential decay); if r = −1, the terms alternate between a and −a; and if r < −1, the sequence alternates in sign while growing in magnitude. The nth term formula is: aₙ = a₁ × rⁿ⁻¹.
The sum of a finite geometric sequence of n terms is: Sₙ = a₁ × (1 − rⁿ) / (1 − r) for r ≠ 1. When |r| < 1, the infinite series converges to a finite sum: S∞ = a₁ / (1 − r). This remarkable result — that an infinite series can have a finite sum — is one of the foundational results of mathematical analysis. The classic illustration is Zeno's paradox of Achilles and the tortoise, where the infinite sum ½ + ¼ + ⅛ + … = 1, showing that infinitely many steps can cover a finite distance in finite time.
Geometric sequences appear throughout mathematics, finance, and science. Compound interest is a geometric sequence — each year's balance is the previous year's multiplied by (1 + r). Radioactive decay is a geometric sequence in discrete time steps — each period retains a fixed fraction of the remaining material. Population growth, bacterial doubling, and depreciation on a declining-balance schedule all follow geometric progressions. This calculator computes the nth term, partial sums, infinite series sum (when convergent), and lists the first n terms for any geometric sequence defined by its first term and common ratio.
Geometric Sequence Formulas
How the Geometric Sequence Calculator Works
Formula, assumptions, and calculation steps for this math tool.
Methodology
Math calculators apply the relevant arithmetic, algebraic, geometric, or numeric rule to the values entered and simplify the result where possible.
Calculation Steps
- Read the values and operation selected.
- Normalize signs, decimals, fractions, or units if needed.
- Apply the mathematical rule or formula.
- Format the answer and any intermediate values for checking.
Assumptions and Limits
- Inputs must be within the supported domain of the operation.
- Decimal answers may be rounded for readability.
- Symbolic simplification is limited to the calculator scope.
Frequently Asked Questions
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 + …
Real-World Applications
Common Mistakes
Geometric Sequence Formulas Quick Reference
| Formula | Expression | Condition |
|---|---|---|
| nth Term | aₙ = a₁ × r^(n−1) | Any r |
| Common Ratio | r = aₙ / aₙ₋₁ | n ≥ 2 |
| Sum of n Terms | Sₙ = a₁(1−rⁿ)/(1−r) | r ≠ 1 |
| Sum (r = 1) | Sₙ = n × a₁ | r = 1 only |
| Infinite Sum | S∞ = a₁ / (1−r) | |r| < 1 only |
| Geometric Mean | G = √(a × b) | Of two terms a, b |
References
- Stewart, James, Redlin, Lothar, and Watson, Saleem. Precalculus: Mathematics for Calculus. Cengage, 2016.
- Larson, Ron and Hostetler, Robert P. Algebra and Trigonometry. Cengage, 2016.
- Rudin, Walter. Principles of Mathematical Analysis. McGraw-Hill, 1976.
- Khan Academy. Geometric Sequences and Series. Khan Academy, 2024.
- National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. NCTM, 2000.
Related Calculators
Browse all Math calculators →Arithmetic Sequence Calculator
Calculate any term, sum, or common difference of an arithmetic sequence.
Sequence Calculator
Identify and calculate terms of number sequences including arithmetic, geometric, and Fibonacci.
Exponent Calculator
Calculate exponents, powers, and scientific notation instantly.