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Geometric Sequence Calculator

Find the nth term, finite sum, and infinite sum of a geometric sequence from its first term and common ratio.

Nth Term (aₙ = a₁ · rⁿ⁻¹)
486
Finite Sum (Sₙ)
728
Infinite Sum (S_∞)
Diverges (|r| ≥ 1)
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Calculation Guide & Reference

Geometric Sequence Mathematics

Geometric sequence calculator finds the nth term, finite sum, and infinite sum from the first term and common ratio.

Standardized Mathematical Formula
aₙ = a₁ × r^(n−1)

In a geometric sequence, each term is multiplied by a fixed common ratio r. If |r| < 1, the infinite sum converges to a finite value; otherwise it diverges to infinity.

Variables:
a₁:First term
r:Common ratio between consecutive terms
n:Term number
How It Works (Step-by-Step)
  • 1Enter the first term, common ratio, and term number.
  • 2View the nth term, the finite sum of n terms, and the infinite sum (if it converges).
Real-World Numerical Example
First Term 2, Ratio 3, 6th Term

A geometric sequence starts at 2 with a common ratio of 3. Find the 6th term and the sum of the first 6 terms.

•6th term: 2 × 3⁵ = 2 × 243 = 486.
•Sum: 2 × (1 − 3⁶) / (1 − 3) = 2 × (−728) / (−2) = 728.
•Infinite sum: diverges, since |r| = 3 is not less than 1.
Result: The 6th term is 486, the sum of the first 6 terms is 728, and the infinite sum diverges.
Calculation Best Practices & Tips
Examine the discriminant (b² - 4ac) before solving quadratics to instantly know if roots are real, distinct, repeated, or complex.
In statistical analysis, always check both the mean and median to identify if data distribution is skewed by outliers.
When calculating percentage changes, ensure you divide by the original starting baseline value: ((New - Old) / Old) * 100.

Frequently Asked Questions (FAQ)

A sequence where each term is the previous term multiplied by a constant ratio (r). Example: 2, 6, 18, 54... has r = 3.

Only when the absolute value of the common ratio is less than 1 (|r| < 1). In that case, the sum converges to a₁ / (1 − r); otherwise the terms grow without bound and the sum diverges.

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