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 CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 5/5 declared partitions (arithmetic, geometric, fibonacci, linear-recurrence, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
a_n and S_n for an integer geometric progression, with an engine-linked stem figure of a_k through n.
Scope
The Result Card includes an engine-linked stem figure of a_k (marker is published term; at most 16 indices). Same closed form as REST/MCP. Terms array is not a REST field. Huge BigInt terms omit the figure.
Verification
Engine tested · Specification checked · v1.0.1 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.1 · CVP protocol 1.0.0-proposed · Evidence 2026-10-07.signed-bigint-publication
Verification revision
2026-10-07.signed-bigint-publication · 3/3 property · digest 0c1fda7d6011
Legacy regression
12/12 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production STALE · Overall CVP state: VERIFIED_STALE under CVP 1.0.0-proposed
CVP status
STALE · Capability production binding: STALE · Site report: STALE · Evidence changed after the last successful production attestation. Re-attestation required.
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
8/8 golden · 3/3 CVP boundary · 6/6 invalid · 3/3 property · 5/5 cross-interface · 1/1 cross-calculator · 6/6 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
4 legacy golden · 2 legacy boundary · legacy regression suite · 8/8 oracle-backed golden · 6/6 invalid · Artifact integrity PASS
STALE · Build schema 1.0.1 matches page · production binding STALE · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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. Term index n ≥ 1, maximum 1000. r = 0 gives a₁ = a and aₙ = 0 for n > 1.

How to use

1

Enter a, r, and n

n is the term index, starting at 1.

2

Read a_n and S_n

The figure shows a stem plot of a_k through 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 1.0.1 · 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, with an engine-linked stem figure of a_k through n.
Inputs
  • a
  • r
  • n
Outputs
  • term
  • sum
Formula
a_n = a r^{n−1}
Assumptions
  • The Result Card includes an engine-linked stem figure of a_k (marker is published term; at most 16 indices). Same closed form as REST/MCP. Terms array is not a REST field. Huge BigInt terms omit the figure.
  • 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 6/6 · View evidence

  • a=3 r=2 n=4 → term=24 sum=45
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.1

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. ›

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.