Geometric sequence nth term calculator

Definition

A geometric sequence (geometric progression) is a number sequence (aₙ) in which the ratio of each term to the previous one is constant whenever the previous term is nonzero. That constant is the common ratio, usually written q (many US textbooks use r). Recursive form: given a₁, aₖ₊₁ = q · aₖ for k ≥ 1. Closed form of the nth term: aₙ = a₁ · qⁿ⁻¹ for n = 1, 2, 3, …

When n = 1, q⁰ = 1, so a₁ returns unchanged. The next values are a₂ = a₁ · q and a₃ = a₁ · q². This calculator implements the closed form aₙ = a₁ · qⁿ⁻¹ with the index reduced by n = max(1, floor(n)).

A geometric sequence is multiplicative: each step multiplies by the same factor. That distinguishes it from an arithmetic sequence, where each step adds a constant difference r. In exponential form, aₙ = a₁ · e^{(n−1)·ln q} for q > 0 — that identity links the progression to the exponential function, but this calculator evaluates qⁿ⁻¹ directly, without a logarithm.

If you know two terms aᵢ and aⱼ (i < j, aᵢ ≠ 0), the ratio satisfies q^{j−i} = aⱼ/aᵢ. This page does not solve for q or n — it takes a₁, q and n as inputs and returns aₙ.

Historical and conceptual context

Geometric progressions appear in ancient computational texts (doubling, halving, proportional “percent” ideas). Euclid’s Elements develops proportion theory that later algebra writes as powers of a ratio. In secondary school, geometric sequences sit beside arithmetic ones and before geometric series.

Applications include compound interest (capital after n periods at a fixed rate), population growth with a constant factor, discrete radioactive decay models, cell division, and power series with |q| < 1. The infinite sum S = a₁/(1−q) for |q| < 1 belongs with series sums — here we compute one term.

The name “geometric” ties to the geometric mean: positive terms satisfy aₖ = √(aₖ₋₁ · aₖ₊₁). That identity is not a separate model on this page, but it explains the historical name. In discrete finance the factor (1+i) plays the role of q; in biology q = 2 describes ideal generation doubling.

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 = a₁(1−qⁿ)/(1−q) for q ≠ 1 lives on the separate sum calculator.

Formal properties

Domain of this calculator: finite real a₁ and q; finite input n; n_eff = max(1, ⌊n⌋). The result aₙ = a₁ · q^(n_eff−1) must be a finite floating-point number. With q < 0 and a non-integer exponent, general mathematics may enter the complex plane — here the exponent n_eff−1 is always a nonnegative integer, so the result stays real.

With q = 1, aₙ = a₁ for every n. With q = −1, terms alternate ±a₁. When |q| > 1, |aₙ| grows exponentially in n; when 0 < |q| < 1, the magnitude shrinks toward zero. With q = 0 and n_eff > 1, aₙ = 0 for finite a₁.

Ratio identity: if aₖ ≠ 0, then aₖ₊₁/aₖ = q. Scaling: (c·aₙ) keeps the same q. Reciprocals (when aₙ ≠ 0) have common ratio 1/q. The product of the first n positive terms is a₁ⁿ · q^{0+1+…+(n−1)} = a₁ⁿ · q^{n(n−1)/2} — that is a combinatorial aside, not this page’s output.

Monotonicity for a₁ > 0 and q > 1: increasing; for 0 < q < 1: decreasing toward zero. For a₁ < 0, signs and monotonicity flip with a₁. Oscillation appears when q < 0.

Full derivation

Step 1. Take a₁ and q. From the recurrence, a₂ = q·a₁, a₃ = q·a₂ = q²·a₁, a₄ = q³·a₁. By induction, aₙ = qⁿ⁻¹ · a₁.

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

Step 3. Compute p = q^(n_eff−1), then aₙ = a₁ · p.

Step 4. Example 2, 3, 4: p = 3³ = 27, a₄ = 2·27 = 54. Example 1, 2, 5: p = 2⁴ = 16, a₅ = 16. Example 5, 0.5, 3: p = 0.25, a₃ = 1.25.

The panel also shows aₙ₋₁ (when n_eff > 1) and aₙ₊₁ = aₙ · q, so the multiplicative step is visible both ways.

Special cases

2, 3, 4 → 54. 1, 2, 5 → 16. 5, 0.5, 3 → 1.25. 3, −2, 4 → −24. 10, 1, 7 → 10 (constant). 8, 0, 3 → 0. 2, 3, 4.9 → 54 (floor). 2, 3, 0 → a₁ = 2 (n → 1).

Very large |q| and n may yield Infinity in floating-point arithmetic — the calculator reports an out-of-range error instead of a silent infinity in the primary field.

Numerical notes

Computation: q^(n_eff−1) · a₁ in the browser’s floating-point arithmetic. For ordinary school examples the displayed value matches the decimal intent; extreme powers 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: arithmetic sequence term, geometric series sum, compound interest, geometric mean of two numbers, power aⁿ. An infinite geometric series needs |q| < 1 — a separate topic.

Sources: Wikipedia — Geometric progression; MathWorld — Geometric Progression; standard school algebra aₙ = a₁·qⁿ⁻¹. Educational tool.