Geometric Sequence Calculator
Find the nth term, partial sum, and infinite sum of any geometric sequence. Determine convergence and generate terms for any ratio.
What Is the Geometric Sequence Calculator?
A geometric sequence is a list of numbers where each term is multiplied by a constant ratio r to get the next. When |r| < 1, the terms shrink toward zero and the infinite series converges to a finite sum. When |r| > 1, the terms grow without bound and the infinite series diverges. This calculator computes the nth term, partial sum, and — when applicable — the infinite sum.
Formula
How to Use
Enter the first term, common ratio, and number of terms. The calculator shows the nth term, the partial sum of n terms, and the infinite sum (if |r| < 1). Up to 20 terms are listed. Try a ratio between −1 and 1 to see convergence.
Example Calculation
First term = 4, ratio = 0.5, n = 8: a₈ = 4 × 0.5^7 = 0.03125. S₈ = 4 × (1 − 0.5^8) / (1 − 0.5) ≈ 7.969. S∞ = 4 / (1 − 0.5) = 8.
Understanding Geometric Sequence
Geometric sequences are one of the most important structures in mathematics and natural science. Exponential growth and decay — the basis of compound interest, epidemic modelling, radioactive half-lives, and population dynamics — are all geometric sequences in continuous form.
The infinite geometric series formula S∞ = a/(1−r) is one of the most-used results in calculus and analysis. It underlies the derivation of the formula for present value of perpetuities in finance, the Basel problem in number theory, and numerous power series expansions.
Zeno's paradox of Achilles and the tortoise is resolved by recognising that an infinite number of diminishing steps can sum to a finite distance — precisely the convergent geometric series with r < 1.
Frequently Asked Questions
When does a geometric series converge?
A geometric series converges (has a finite infinite sum) when and only when the absolute value of the common ratio |r| < 1. The infinite sum is a₁ / (1 − r).
What are real-world examples of geometric sequences?
Compound interest growth (r = 1 + interest rate), radioactive decay (r < 1), population growth models, bouncing ball height, digital signal attenuation, and Moore's Law (transistor count doubling roughly every 2 years).
What happens when r = 1?
When r = 1, every term equals the first term — it is a constant sequence. The partial sum formula divides by zero, so Sₙ simply equals n × a₁. The infinite sum diverges.
Can r be negative?
Yes. A negative ratio produces an alternating sequence: positive, negative, positive, … For example, a₁ = 1, r = −1/2: 1, −0.5, 0.25, −0.125, … This still converges when |r| < 1.
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