Arithmetic sequence nth term calculator

Definition

An arithmetic sequence (arithmetic progression) is a number sequence (aₙ) in which the difference between each term and the previous one is constant. That constant is the common difference, usually written r (many US textbooks use d). Recursive form: given a₁, aₖ₊₁ = aₖ + r for k ≥ 1. Closed form of the nth term: aₙ = a₁ + (n−1)·r for n = 1, 2, 3, …

For n = 1, (1−1)·r = 0, so a₁ returns unchanged. For n = 2: a₂ = a₁ + r. For n = 3: a₃ = a₁ + 2r. This calculator implements the closed form aₙ = a₁ + (n−1)·r with the index reduced by n = max(1, floor(n)).

An arithmetic sequence is additive: each step adds the same amount. That distinguishes it from a geometric sequence, where each step multiplies by a constant ratio q. The points (k, aₖ) lie on a straight line of slope r — hence the “linear” discrete growth picture.

If you know two terms aᵢ and aⱼ (i < j), the difference satisfies r = (aⱼ − aᵢ)/(j − i). This page does not solve for r or n — it takes a₁, r and n as inputs and returns aₙ.

Historical and conceptual context

Arithmetic progressions belong to the oldest computational patterns: equal increments, house numbers in a row, regular deposits. Euclid’s Elements develops means and proportions; modern algebra writes the same idea as a formula linear in n. In secondary school, arithmetic sequences sit beside geometric ones and before series sums.

Applications include instalments that decline by a fixed principal share, temperatures at equal time gaps, house numbers on one side of a street, and discrete uniform motion xₖ = x₀ + v·Δt·(k−1). The sum Sₙ = n·(a₁ + aₙ)/2 belongs with series sums — here we compute one term.

The name “arithmetic” ties to the arithmetic mean: each interior term is the mean of its neighbours, aₖ = (aₖ₋₁ + aₖ₊₁)/2. That identity is not a separate model on this page, but it explains the historical name.

Distinguish a finite sequence (the list a₁…aₙ) from a series (a sum of terms). This calculator belongs to the first world: it points to position n. The finite sum Sₙ lives on the separate arithmetic-series sum calculator.

Formal properties

Domain of this calculator: finite real a₁ and r; finite input n; n_eff = max(1, ⌊n⌋). The result aₙ = a₁ + (n_eff−1)·r must be a finite floating-point number. There is no exponentiation — only multiply and add — so there is no complex-branch risk like a negative base raised to a non-integer power.

If r = 0, then aₙ = a₁ for every n (constant sequence). If r > 0 the sequence is strictly increasing; if r < 0 it is strictly decreasing. Scaling: (c·aₙ) has common difference c·r. Shifting the index does not change r.

Difference identity: aₖ₊₁ − aₖ = r. The arithmetic mean of the first n terms is (a₁ + aₙ)/2. The product of the first n terms has no simple elementary closed form in general — unlike a geometric sequence.

Monotonicity follows only from the sign of r. Oscillation does not appear for a fixed real r (unlike a negative ratio q in a geometric sequence).

Full derivation

Step 1. Take a₁ and r. From the recurrence, a₂ = a₁ + r, a₃ = a₂ + r = a₁ + 2r, a₄ = a₁ + 3r. By induction, aₙ = a₁ + (n−1)·r.

Step 2. On this page read fields a₁, r, n. Set n_eff = max(1, floor(n)).

Step 3. Compute the step s = (n_eff−1)·r, then aₙ = a₁ + s.

Step 4. Example 1, 2, 5: s = 4·2 = 8, a₅ = 1+8 = 9. Example 3, 5, 4: s = 3·5 = 15, a₄ = 18. Example 10, −3, 6: s = 5·(−3) = −15, a₆ = −5.

The panel also shows aₙ₋₁ (when n_eff > 1) and aₙ₊₁ = aₙ + r so the additive step is visible both ways.

Special cases

1, 2, 5 → 9. 3, 5, 4 → 18. 10, −3, 6 → −5. 2, 0.5, 4 → 3.5. 7, 0, 5 → 7 (constant). 1, 2, 4.9 → 7 (floor to 4). 0, 5, 3 → 10. 1, 2, 0 → a₁ = 1 (n → 1).

Very large |r| and n may yield Infinity in floating-point arithmetic — the calculator reports an out-of-range error.

Numerical notes

Computation: a1 + (nEff − 1) * r in the browser’s floating-point arithmetic. For ordinary school examples the displayed value matches the decimal intent; extreme products meet floating-point rounding. This is an educational tool, not a CAS.

The calculator accepts a comma or a period as the decimal separator. Formatting follows the page language, without a forced six-decimal trim — with no extra rounding.

Bars illustrate |aₖ| for the first min(n, 8) terms; when n > 8 the full aₙ still appears in the panel.

Related notions and sources

Related: geometric sequence term, arithmetic series sum, arithmetic mean, linear function value, linear interpolation. An infinite arithmetic series diverges when r ≠ 0 — a separate analysis topic.

Sources: Wikipedia — Arithmetic progression; MathWorld — Arithmetic Progression; standard school algebra aₙ = a₁+(n−1)r. Educational tool.