Arithmetic Sequence Formula

Variables

• aₙ: nth term
• a₁: first term
• n: term number
• d: common difference

How to use this formula

Finds the nth term of a sequence with a constant difference.

Important notes

• The difference between consecutive terms must be constant.
• Term numbering begins at n = 1.

Quick example

For 3, 7, 11, … the 10th term is 39.

Applicability, worked calculation, and verification

Domain and applicability

Applies to sequences with a constant additive difference between consecutive terms and integer term index n ≥ 1.

Units

• a₁, d, and aₙ use the same units
• n is dimensionless

Assumptions and domain checks

• The difference between consecutive terms must be constant.
• For the Arithmetic Sequence Formula, all variables must lie in the stated domain, and every denominator or inverse operation must be defined.

Do not use this formula when

• Do not use Arithmetic Sequence Formula when its variable definitions, domain restrictions, or structural assumptions differ from the problem.

Boundary and special cases

• For Arithmetic Sequence Formula, check zero, negative, and extreme input values before relying on the result.
• When using Arithmetic Sequence Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.

Equivalent and alternative forms

• Keep the canonical LaTeX form a_n=a_1+(n-1)d for copying; rearrange only after preserving equivalence and domain restrictions.

Worked example

For a sequence with first term 3 and common difference 4, the tenth term is a₁₀ = 3 + (10 − 1)4 = 39.

1. Identify a₁ = 3, d = 4, and n = 10.
2. Compute n − 1 = 9 intervals from the first term.
3. Multiply the common difference: 9 × 4 = 36.
4. Add the first term to obtain a₁₀ = 39.

Independent verification

Listing the terms 3, 7, 11, … shows that adding 4 nine times reaches 39 at the tenth position.

Common mistakes

• Before substituting values into Arithmetic Sequence Formula, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
• For the Arithmetic Sequence Formula, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.

Review and verification

Last reviewed: 2026-07-26

Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Arithmetic Sequence Formula.

Verified against: OpenStax Algebra and Trigonometry

Automated quality check: Passed the core formula indexing gate.

Formula references

• Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.

Frequently asked questions