HomeCalculatorsMathNumber Theory & Discrete MathArithmetic Sequence Calculator
Math calculator

Arithmetic Sequence Calculator

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

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · 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 arithmetic 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.
Verification
Engine tested · Source checked · v1.0.1 · CVP VERIFIED · 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 CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
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
Sources
Sources
Evidence
4 legacy golden · 2 legacy boundary · legacy regression suite · 8/8 oracle-backed golden · 6/6 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.1 matches · Semantic contract ✓ · Production attested · 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 + (n−1)d
SumS_n = n/2 · (2a + (n−1)d)
iIntegers only. Exact BigInt. Term index n ≥ 1, maximum 1000. Engine id is sequence — never the add/sub arithmetic engine.

How to use

1

Enter a, d, 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

2, 5, 8, …

a=2 d=3 n=5

a_5=14
14

Arithmetic Sequence calculator specification

Version 1.0.1 · Engine tested

Calculation status

Review policy · Evidence

Definition
a_n = a + (n−1)d and S_n = n/2 · (2a + (n−1)d). This page is mode=arithmetic on the sequence engine. Not the add/sub arithmetic calculator.
What it calculates
a_n and S_n for an integer arithmetic progression, with an engine-linked stem figure of a_k through n.
Inputs
  • a
  • d
  • n
Outputs
  • term
  • sum
Formula
a_n = a + (n−1)d
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.
  • Integers
  • n ≥ 1
  • Not geometric / Fibonacci / Pascal
Units
  • dimensionless
Boundary conditions
  • n=0 → VALUE_MUST_BE_POSITIVE
  • n>1000 → VALUE_ABOVE_MAX
Example
a=2 d=3 n=5 → 14
Validation cases

1 published on this page · 12/12 tests · Production surface contract 6/6 · View evidence

  • a=2 d=3 n=5 → term=14 sum=40
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
  • Margaret Stevens — The Maths Workbook (University of Oxford) — Chapter 3, §2: Arithmetic and Geometric Sequences and Series
    Supports: Arithmetic terms and sums from first term and common difference, geometric terms and finite sums from first term and common ratio, including derivations. At ratio 1 the finite geometric sum is n times the first term; it is the removable special case of the quoted expression. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
Calculation version
1.0.1

Background

Interpretation and common distinctions.

What does this arithmetic sequence calculator compute?

a_n = a + (n−1)d and S_n = n/2 · (2a + (n−1)d).

The published formula or method is a_n = a + (n−1)d. The following references support the stated mathematics or method:

  • Margaret Stevens — The Maths Workbook (University of Oxford): Arithmetic terms and sums from first term and common difference, geometric terms and finite sums from first term and common ratio, including derivations. At ratio 1 the finite geometric sum is n times the first term; it is the removable special case of the quoted expression. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Check the sequence type, starting-index convention, and finite-term count. Preserve complete terms and exact integers where the result provides them.

The worked example uses a=2 d=3 n=5 with a_5=14. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.sequences v1.0.1 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

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

→ 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. › → Geometric Sequence Calculator Nth term and partial sum of an integer geometric sequence. Same engine as Sequence. Not arithmetic. Runs locally. › → 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 the add/subtract calculator?

No. The engine id is sequence. The add/sub engine remains arithmetic and must not be aliased here.

Is this a second sequence solver?

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