Geometric Sequence Calculator

About the Geometric Sequence Calculator

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).

When This Page Helps

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.

How to Use the Inputs

1. Select a calculation mode: nth term, sum, common ratio, or infinite sum.
2. Enter the first term (a₁) of your geometric sequence.
3. Provide the common ratio (r) — values > 1 give growth, 0 < r < 1 gives decay.
4. Enter the number of terms (n) for finite calculations.
5. Try presets like Powers of 2 or Half-life decay to explore common patterns.
6. Review the output cards and scroll to the growth/decay visualization.
7. Check the table for term-by-term values and running partial sums.

Example Calculation

Tips & Best Practices

• When |r| < 1, the sequence converges to 0 and the infinite sum is finite.
• Negative r values cause alternating positive/negative terms.
• Compound interest at rate p% is a geometric sequence with r = 1 + p/100.
• The geometric mean of two numbers a and b is √(a·b) — the middle term.
• Every geometric sequence with r > 0 can be written as a₁·e^(kn) for exponential modeling.

Exponential Growth vs. Exponential Decay

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.

Infinite Geometric Series and Convergence

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).

Geometric Sequences in Finance and Science

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.

Frequently Asked Questions

• What is a geometric sequence?

  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.

• When does the infinite sum converge?

  Only when |r| < 1. The sum S∞ = a₁/(1 − r). If |r| ≥ 1, the series diverges to infinity.

• Can r be negative?

  Yes. A negative ratio produces alternating signs: 2, −6, 18, −54, … (r = −3).

• How does this differ from arithmetic sequences?

  Arithmetic sequences add a constant each step (linear); geometric sequences multiply by a constant (exponential).

• What are real-world examples?

  Compound interest, bacterial growth, radioactive decay, mortgage payments, fractal patterns, and signal loss over distance.

• What happens when r = 1?

  Every term equals a₁ — it's a constant sequence. The sum is simply n · a₁.