Geometric Sequence, Finite Sum & Infinite Series Calculator

Evaluate geometric progressions a_n = a₁r^(n−1) and infinite convergence a₁/(1−r).

Finite Geometric Sum (S_n)
199.219
nth Term (a_n) 0.781
Infinite Series Sum S_∞ (|r|<1) 200
Ratio Power r^n 0.004
Inputs
First term (a₁).
Common ratio (r).
Number of terms (n).
Results & Graph
Finite Geometric Sum (S_n)
199.219
nth Term (a_n)
0.781
Infinite Series Sum S_∞ (|r|<1)
200
Ratio Power r^n
0.004

Finite Geometric Sum (S_n)

Graph 1 • First Term A1 Sensitivity Recalculates the domain equation across 50% to 150% of the primary input while holding other parameters constant.
0 161.4 322.7 50%: 99.609 99.6 50% 75%: 149.414 149.4 75% 100%: 199.219 199.2 100% 125%: 249.023 249.0 125% 150%: 298.828 298.8 150%
Graph 2 • Primary & Supporting Metric Comparison Compares the primary output alongside supporting domain benchmarks.
Finite Geometric Sum (S_n) 199.219 nth Term (a_n) 0.781 Infinite Series Sum S_∞ (|r|<1) 200 Ratio Power r^n 0.004
Graph 3 • Exact Output Table & Units Displays each calculated output in its exact domain unit.
Finite Geometric Sum (S_n)199.219nth Term (a_n)0.781Infinite Series Sum S_∞ (|r|<1)200Ratio Power r^n0.004
Summary: With a₁ = 100 and r = 0.5, the 8th term is 0.7813 and finite sum S_8 = 199.2188.
Summary: With a₁ = 100 and r = 0.5, the 8th term is 0.7813 and finite sum S_8 = 199.2188.

Formula & Mathematical Methodology: Geometric Series Mathematical Identity

an = a1 × rn−1
Sn = a11 − rn1 − r
S_∞ = a1/(1 − r)
a_n = a_1 × r^(n−1); S_n = a_1(1 − r^n)/(1 − r); S_∞ = a_1/(1 − r)
  • firstTermA1: First Term A1 (numeric parameter)
  • commonRatioR: Common Ratio R (numeric parameter)
  • termCountN: Term Count N (numeric parameter)

Rounding: All calculations run in double-precision IEEE-754 arithmetic; displayed outputs round to up to 4 decimal places.

Worked Example: Worked Mathematical Verification: Geometric Series

Evaluate geometric progressions a_n = a₁r^(n−1) and infinite convergence a₁/(1−r).

  1. Verify the domain constraints of the input parameters.
  2. Apply the governing mathematical identity: a_n = a_1 × r^(n−1); S_n = a_1(1 − r^n)/(1 − r); S_∞ = a_1/(1 − r).
  3. Cross-check the primary value against the supporting invariants and coordinate graph.
Result: Evaluates Geometric Series via a_n = a_1 × r^(n−1); S_n = a_1(1 − r^n)/(1 − r); S_∞ = a_1/(1 − r).

Frequently Asked Questions

What mathematical formula does the Geometric Sequence, Finite Sum & Infinite Series Calculator use?

It directly evaluates a_n = a_1 × r^(n−1); S_n = a_1(1 − r^n)/(1 − r); S_∞ = a_1/(1 − r) in double-precision floating-point arithmetic and updates all supporting invariants and graphs live.

How can I compare different parameter sets?

Click Compare in the results toolbar to pin up to three parameter combinations side by side.

Can I export the mathematical report or share a link?

Yes. Use the Share Link, Copy Summary, or Markdown/JSON export buttons in the results panel.

Mathematics & Statistics • Sequences & Series Guide

How to Use the Geometric Sequence, Finite Sum & Infinite Series Calculator: Practical Guide & Visual Analysis

Calculate the nth term a_n, finite geometric sum S_n, and convergent infinite series sum S_∞ (when |r| < 1). Includes textbook mathematical notation, step-by-step verification, and three custom SVG visualizations. Whether you are researching geometric sequence, finite sum & infinite series calculator, geometric series, geometric sequence infinite series, sequences & series calculator, this interactive tool helps you calculate the nth term a_n, finite geometric sum S_n, and convergent infinite series sum S_∞ (when |r| < 1).

Interpreting the 3 Custom Visual Graphs & Key Outputs

As you adjust First Term A1, Common Ratio R, Term Count N, the calculator immediately computes Primary exact or numerical solution, Supporting identities & invariants, 3-Panel mathematical SVG plots and updates all three domain-specific SVG charts in real time. Instead of relying on a single static figure, you can inspect the primary domain diagram, component breakdown, and multi-point trajectory curve to see how changing your inputs shifts the final result. Use the Compare (up to 3 scenarios) button to evaluate side-by-side variations before making a decision.

Real-World Benchmark & Planning Tips (Universal Mathematical & Statistical Standard (ISO 80000-2))

In our benchmark scenario (Worked Mathematical Verification: Geometric Series), Evaluates Geometric Series via a_n = a_1 × r^(n−1); S_n = a_1(1 − r^n)/(1 − r); S_∞ = a_1/(1 − r). Keep in mind that uses standard Euclidean geometry, real/complex algebra, and NIST statistical definitions. For related calculations in Mathematics & Statistics, explore Scientific Calculator, Statistics & Standard Deviation Calculator, Geometry Calculator (Right Triangle, Circle & Cylinder).