EVERYDAY MATH

Arithmetic sequence calculator

Calculate an arithmetic sequence term and finite sum from a first term and common difference, or recover the sequence from two known terms.

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Use a first term and common difference, or recover the sequence from two known terms. Term numbers must be positive whole numbers.

Known informationFirst term a₁Common difference dTarget term number n

Arithmetic sequence result

First term a₁: 3
Common difference d: 2
Term a₁₀: 21
Sum S₁₀: 120

First 10 terms

a₁ = 3
a₂ = 5
a₃ = 7
a₄ = 9
a₅ = 11
a₆ = 13
a₇ = 15
a₈ = 17
a₉ = 19
a₁₀ = 21

Arithmetic sequences use one constant difference. This tool does not model geometric, recursive, random, or irregular sequences.

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Verification and calculation details

Purpose: Arithmetic sequence calculator
Variables and units: Use the labelled fields and units shown in the calculator. Results use those units unless a conversion is stated.
Formula and method: The exact formula and worked example are shown below. Deterministic formula executed locally in the browser.
Assumptions: Only entered values and stated assumptions are included. Local fees, rules and professional requirements are excluded unless explicitly shown.
Engine version: 1.0.0
Production verification baseline: Release 63 · source 998803f
Verification run: release-v63-2026-09-20
Last verified: 2026-09-20 — Automated release checks passing

Read the Calculation Standard · Report a calculation or source issue

How this calculator works

For first term a₁, common difference d and positive integer n, aₙ=a₁+(n−1)d and Sₙ=n(a₁+aₙ)/2. From known terms aₘ and aₙ at different indices, d=(aₙ−aₘ)/(n−m), then a₁=aₘ−(m−1)d.

Variables and inputs

Known information: Required numeric input. allowlisted selection: first term and difference, or two known terms.
First term or first known term value: Required numeric input. finite; absolute value no more than 1 trillion.
Common difference: Optional numeric input. finite; absolute value no more than 1 trillion.
First known term number: Optional numeric input. positive whole number no more than 1,000,000; different from the second known term number.
Second known term number: Optional numeric input. positive whole number no more than 1,000,000; different from the first known term number.
Second known term value: Optional numeric input. finite; absolute value no more than 1 trillion.
Target term number: Required numeric input. positive whole number no more than 1,000,000.

How to interpret the result

The calculator returns first term, common difference, selected nth term, finite sum through the selected term, first 12 terms or fewer. Read each output against the assumptions you entered, compare alternatives one variable at a time, and use the result as an estimate rather than a promise.

Common mistakes to avoid

Mixing units or entering a rate as a decimal when the field requests a percentage.
Rounding intermediate values instead of the final displayed result.
Leaving out a cost, timing rule or local requirement listed in the assumptions below.

Practical uses

Use the arithmetic sequence calculator to check a real scenario, compare a changed assumption and explain the arithmetic before relying on a quote, specification or decision.

Worked example

For a₁=3, d=2 and n=10, a₁₀=3+(10−1)×2=21 and S₁₀=10×(3+21)/2=120. The same sequence can be recovered from a₁=3 and a₄=9 because d=(9−3)/(4−1)=2.

Assumptions and limitations

the difference between every adjacent pair is constant
known term indices and the target index are positive integers
known term values are exact inputs
the finite sum starts at the first term and ends at the selected target term

does not model geometric, recursive, random or irregular sequences
does not infer a pattern from a list of observations
large indices can magnify rounding in decimal inputs
displayed decimals are rounded after full-precision calculation

Important notes

Results are estimates based on the values entered. Confirm rates, prices, units and professional requirements that apply in your location. Review the Answer Atlas Calculation Standard and corrections policy for formula, privacy and correction rules.

Frequently asked questions

What makes a sequence arithmetic?

An arithmetic sequence has the same difference between each pair of consecutive terms.

Can I find the common difference from two terms?

Yes. Give two term values at different positive term numbers. The calculator divides the change in value by the change in term number and then recovers the first term.

What does the finite sum include?

Sₙ adds every term from a₁ through the selected term aₙ. The on-page table is capped at 12 rows, but the sum uses the full selected n.