Geometric Sequence Calculator

aₙ = a₁ × r^(n−1). Find any term of a geometric sequence.

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Result

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How to use this calculator

  • Enter a₁, ratio r, and n.

About this calculator

Geometric sequence: each term is the previous × constant ratio r. Formula aₙ = a₁ × r^(n−1). Sum: Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1. Examples: doubling (r=2), halving (r=½), compound interest (r = 1+rate), bouncing-ball heights (r<1), nuclear chain reactions. Infinite sum converges if |r| < 1: S∞ = a₁/(1−r).

Frequently asked

r = 1 case?+
All terms equal a₁; sum = n × a₁. Standard formula has 0/0; this calc handles separately.
Negative r?+
Alternating sign. r = −1: alternates +a₁, −a₁, +a₁, ...
|r| < 1 infinite sum?+
Converges to a₁/(1−r). a₁=10, r=0.5: 10 + 5 + 2.5 + ... = 20.
Compound interest?+
Geometric with r = 1+rate. $1000 at 5% over 30 yr = 1000 × 1.05^30 = $4322.
Difference from arithmetic?+
Arithmetic adds; geometric multiplies. Linear vs exponential growth.

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Sₙ = a₁(1 − rⁿ) / (1 − r) for r ≠ 1. Infinite: S∞ = a₁/(1−r) if |r|<1.
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