What happens if the common ratio r = 1?

If r = 1, every term equals the first term a₁, and the finite sum of n terms is simply S_n = n × a₁.

Geometric Sequence Calculator : Find nth Term (a_n), Finite Sum (S_n) & Infinite Sum (S_∞)

Q: When does an infinite geometric series converge to a finite sum?

An infinite geometric series converges if and only if the absolute value of the common ratio is strictly less than 1 (|r| < 1). In this case, S_∞ = a₁ / (1 - r).

The Calcivo Geometric Sequence Calculator solves geometric progressions (GP). It evaluates the nth term, finite sum of n terms (S_n), and the sum of convergent infinite geometric series (S_∞) when the absolute common ratio |r| < 1.

Interactive Calculator

100% Client-Side Engine

Value of nth Term (a₁₀)

1,536

Finite Sum S₁₀

3,069

Infinite Series Sum (S_∞)

Divergent (|r| ≥ 1)

First 10 Terms

[3, 6, 12, 24, 48, 96, 192, 384, 768, 1536]

Geometric Progression Derivation

1. nth Term Formula: a_n = a_1 × r^{n-1}

a_10 = 3 × (2)^{9} = 1536

2. Finite Sum Formula: S_n = a_1 \frac{1 - r^n}{1 - r} = 3 × (1 - 1024) / (1 - 2) = 3069

3. Infinite series is divergent (|r| ≥ 1), no finite sum exists.

Step-by-Step Examples

Geometric Series with a₁ = 3, r = 2, n = 10

Evaluating doubling progression.

Inputs: First Term (a₁): 3, Common Ratio (r): 2, Position (n): 10

Result: a₁₀ = 1536, S₁₀ = 3069

a₁₀ = 3 × 2⁹ = 3 × 512 = 1536. S₁₀ = 3 × (1 - 1024) / (1 - 2) = 3 × 1023 = 3069.

How to Use This Tool

Enter the initial term (a₁).
Enter the constant common ratio (r).
Enter the number of terms (n).
Click Calculate to see the nth term, finite sum, and infinite series convergence.

Frequently Asked Questions

An infinite geometric series converges if and only if the absolute value of the common ratio is strictly less than 1 (|r| < 1). In this case, S_∞ = a₁ / (1 - r).

Related Calculators & Tools

Explore complementary calculators in this category.

Calculators

Arithmetic Sequence Calculator

Calculate the nth term (a_n = a_1 + (n-1)d), sum of first n terms (S_n), common difference, and generate the arithmetic progression.

Verified Math0ms Latency

Calculators

Sequence Calculator

Identify the pattern in any sequence of numbers. Finds common differences, common ratios, general formulas (a_n), and next predicted terms.

Verified Math0ms Latency

Calculators

Exponent Calculator

Calculate powers and exponents (base^exponent). Supports negative exponents, fractional powers, scientific notation, and reciprocal rules.

Verified Math0ms Latency

Calculators

Percentage Calculator

Calculate what is X% of Y, find percentage increase/decrease, and solve percentage differences instantly.

Verified Math0ms Latency