This arithmetic sequence calculator finds the nth term (aₙ = a₁ + (n−1)d) and the sum of the first n terms of any sequence with a constant difference.

nth term aₙ –
Sum of first n terms Sₙ –
First terms –

The Constant-Step Sequence: aₙ = a₁ + (n − 1)d

Its multiplicative cousin is our Geometric Sequence Calculator. Part of our Math Calculators collection.

Worked Example (Verified)

  • a₁ = 3, d = 5, n = 20 (defaults): a₂₀ = 3 + 19 × 5 = 98 · S₂₀ = 20/2 × (3 + 98) = 1,010 — Gauss’s pairing trick (first + last) × pairs, the formula’s origin story
  • Stacked-seat rows, salary steps, loan balances on simple schedules — anything adding the same amount per step is this sequence wearing a uniform
  • The (n − 1) trap: reaching term n takes n − 1 steps of d, not n — off-by-one is the most common sequence error in existence

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Frequently Asked Questions

What is the formula for the nth term of an arithmetic sequence?

aₙ = a₁ + (n − 1)d, where a₁ is the first term and d the common difference. Example: a₁ = 3, d = 5 → a₂₀ = 3 + 19 × 5 = 98. This calculator evaluates it for any n.

How do you find the sum of an arithmetic sequence?

Sₙ = n/2 × (a₁ + aₙ) — the number of terms times the average of first and last (equivalently n/2 × (2a₁ + (n−1)d)). For the defaults: 20/2 × 101 = 1,010.

How do you find the common difference?

Subtract any term from the next: d = a₂ − a₁. If two non-adjacent terms are known, d = (aₙ − aₘ)/(n − m). A sequence is arithmetic exactly when this difference is constant throughout.

Can the common difference be negative or a fraction?

Yes: d = −2 gives countdowns (30, 28, 26…), d = 0.5 gives half-steps. d = 0 is the constant sequence — legal, boring, and the reason the formula never divides by d.

Arithmetic vs geometric — what’s the difference?

Arithmetic adds a constant (d) each step; geometric multiplies by a constant (r). Growth patterns split exactly there: salary raises by $2,000/yr are arithmetic; populations growing 2%/yr are geometric.

Is this arithmetic sequence calculator free?

Yes — free, instant, no sign-up.