Calculate nth term, finite and infinite sums, common ratio, and visualize exponential growth or decay in geometric sequences.
A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a fixed value called the common ratio. These sequences model exponential growth and decay — from compound interest and population growth to radioactive half-lives and signal attenuation.
This calculator handles four modes: finding the nth term, computing finite sums, determining the common ratio from known endpoints, and evaluating infinite sums for convergent series (where |r| < 1). It generates the full sequence, computes partial sums, and provides a visual bar chart showing the characteristic exponential curve.
The nth term formula is aₙ = a₁ · r^(n−1), and the partial sum is Sₙ = a₁(1 − rⁿ)/(1 − r). When |r| < 1, the infinite sum converges to S∞ = a₁/(1 − r). Understanding geometric sequences is essential in finance (compound interest, annuities), biology (population models), physics (wave behavior, decay), and computer science (algorithm analysis, binary trees).
Geometric sequence calculations involve exponential terms that grow or shrink rapidly — computing r^(n−1) for large n by hand is impractical, and a small rounding error in the ratio compounds through every subsequent term. This calculator handles all four modes (nth term, finite sum, infinite sum, common ratio), generates full sequences with partial sums, identifies growth vs. decay, and visualizes the characteristic exponential curve. It is invaluable for verifying compound interest projections, half-life problems, and convergence questions in calculus.
aₙ = a₁ · r^(n−1) Sₙ = a₁ · (1 − rⁿ) / (1 − r) S∞ = a₁ / (1 − r) when |r| < 1
Result: a₈ = 4,374, S₈ = 6,560
Starting at 2 and multiplying by 3 each time: 2, 6, 18, 54, …, 4374. The 8th term is 2·3⁷ = 4374. The sum is 2·(1−3⁸)/(1−3) = 6560.
The common ratio r determines whether a geometric sequence grows, decays, or oscillates. When r > 1, each term is larger than the last and the sequence models exponential growth — compound interest at rate p% has r = 1 + p/100, bacterial populations that double each hour have r = 2, and inflation compounds geometrically. When 0 < r < 1, the sequence decays toward zero, modeling radioactive half-lives (r = 0.5), signal attenuation, and cooling curves. Negative r values cause alternating signs, producing oscillating sequences that appear in certain electrical circuits and spring-damper systems.
One of the most powerful results in mathematics is that a geometric series with |r| < 1 converges to a finite sum S∞ = a₁/(1 − r), even though infinitely many terms are added. This is the foundation of Zeno's paradox resolution, repeated decimal representations (0.333… = 1/3), and present-value calculations in finance. When |r| ≥ 1, the series diverges — the partial sums grow without bound. The boundary case |r| = 1 is degenerate: the sum is either n·a₁ (r = 1) or oscillates (r = −1).
Compound interest is the most ubiquitous geometric sequence: an initial principal P growing at rate r per period becomes P·rⁿ after n periods. Mortgage amortization, bond pricing, and annuity calculations all rely on geometric series sums. In biology, population models (exponential phase of growth), pharmacokinetics (drug half-life), and genetics (allele frequency changes) follow geometric patterns. In physics, wave amplitude decay, fractal self-similarity, and geometric optics (repeated reflections losing energy) are all geometric sequences in disguise.
A sequence where each term is produced by multiplying the previous term by a constant ratio r. Example: 3, 6, 12, 24 has r = 2.
Only when |r| < 1. The sum S∞ = a₁/(1 − r). If |r| ≥ 1, the series diverges to infinity.
Yes. A negative ratio produces alternating signs: 2, −6, 18, −54, … (r = −3).
Arithmetic sequences add a constant each step (linear); geometric sequences multiply by a constant (exponential). Use this as a practical reminder before finalizing the result.
Compound interest, bacterial growth, radioactive decay, mortgage payments, fractal patterns, and signal loss over distance. Keep this note short and outcome-focused for reuse.
Every term equals a₁ — it's a constant sequence. The sum is simply n · a₁.