HomeCalculatorsMathNumber Theory & Discrete MathGeometric Sequence Calculator
Math calculator

Geometric Sequence Calculator

Nth term and partial sum of an integer geometric sequence. Same engine as Sequence. Not arithmetic. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 12/12 tests · Production surface contract 4/4 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
a_n and S_n for an integer geometric progression.
Scope
Integers
Verification
Engine tested · 12/12 tests · Production surface contract 4/4 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
4 golden · 2 boundary · 6 property · Production surface contract 4/4 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Terma_n = a r^{n−1}
SumS_n = a(r^n − 1)/(r − 1)
iIntegers only. Exact BigInt. n ≥ 1. r=0 gives a_1=a and a_n=0 for n>1. Not arithmetic, not Fibonacci, not Pascal.

How to use

1

Enter a, r, and n

n is the term index, starting at 1.

2

Read a_n and S_n

Larger composition lives at /calc/math/sequence.

Example calculations

Common configurations with formula and result.

ϟ

3, 6, 12, 24

a=3 r=2 n=4

a_4=24
24

Geometric Sequence calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
a_n = a r^{n−1} and S_n = a(r^n−1)/(r−1) (or n a when r=1). This page is mode=geometric on the sequence engine.
What it calculates
a_n and S_n for an integer geometric progression.
Inputs
  • a
  • r
  • n
Outputs
  • term
  • sum
Formula
a_n = a r^{n−1}
Assumptions
  • Integers
  • n ≥ 1
  • Not arithmetic / Fibonacci / Pascal
Units
  • dimensionless
Boundary conditions
  • n=0 → VALUE_MUST_BE_POSITIVE
  • n>1000 → VALUE_ABOVE_MAX
Example
a=3 r=2 n=4 → 24
Validation cases

1 published on this page · 12/12 tests · Production surface contract 4/4 · View evidence

  • a=3 r=2 n=4 → term=24 sum=45
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

Other calculators in this family: Arithmetic Sequence Calculator, Factorial Calculator, Fibonacci Calculator, GCF Calculator, LCM Calculator, Modular Inverse Calculator, Modulo Calculator, Permutation Combination Calculator . Explore all Number Theory & Discrete Math.

→ Arithmetic Sequence Calculator Nth term and partial sum of an integer arithmetic sequence. Same engine as Sequence. Not geometric. Runs locally. › → Factorial Calculator Calculate n! for integers 0 ≤ n ≤ 5000. Fast local exact integers with shareable, machine-readable results. › → Fibonacci Calculator Fibonacci numbers F_n for n ≤ 1000. F_0=0, F_1=1. Same engine as Sequence. Exact BigInt. Runs locally. › → GCF Calculator Find the GCF, GCD, or HCF of two or more integers. Exact BigInt Euclidean fold with factor and prime cross-checks. Free, no sign-up. › → LCM Calculator Find the LCM of two or more integers with exact arithmetic and independent verification. Free, no sign-up. › → Modular Inverse Calculator Modular inverse of a modulo m when gcd(a,m)=1. Same engine as Euclidean remainder. Not a second GCF page. Runs locally. › → Modulo Calculator Integer remainder with a = q·m + r and 0 ≤ r < |m|. Euclidean convention, not JavaScript truncated %. › → Permutation Combination Calculator Exact P(n,k) and C(n,k) for integers 0 ≤ k ≤ n ≤ 1000. Multiplicative BigInt, local-first. Not the factorial calculator. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second sequence solver?

No. It is mode=geometric on math.sequences.

Are non-integer ratios supported?

Not in this seed. Integers only.