EVERYDAY MATH

Geometric sequence calculator

Calculate a geometric sequence term, finite series sum, growth factor and convergent infinite sum from a first term and constant ratio.

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Enter the first term, constant ratio and target term number. The calculator keeps the full formula result while limiting the preview to 12 terms.

First term a₁Common ratio rTarget term number n

Geometric sequence result

Term a₈: 384
Finite sum S₈: 765
Growth factor r⁷: 128
Infinite sum: Does not converge

First 8 terms

a₁ = 3
a₂ = 6
a₃ = 12
a₄ = 24
a₅ = 48
a₆ = 96
a₇ = 192
a₈ = 384

A geometric sequence uses one constant ratio. An infinite sum is shown only when |r| is less than 1. This tool does not infer a pattern from observed values or model arithmetic, recursive or irregular sequences.

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Verification and calculation details

Purpose: Geometric sequence calculator
Variables and units: Use the labelled fields and units shown in the calculator. Results use those units unless a conversion is stated.
Formula and method: The exact formula and worked example are shown below. Deterministic formula executed locally in the browser.
Assumptions: Only entered values and stated assumptions are included. Local fees, rules and professional requirements are excluded unless explicitly shown.
Engine version: 1.0.0
Production verification baseline: Release 63 · source 998803f
Verification run: release-v63-2026-09-20
Last verified: 2026-09-20 — Automated release checks passing

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How this calculator works

The nth term is aₙ=a₁rⁿ⁻¹. The first n terms sum to Sₙ=a₁(1−rⁿ)/(1−r) when r≠1 and Sₙ=na₁ when r=1. The infinite sum is a₁/(1−r) only when |r|<1.

Variables and inputs

First term: Required numeric input. finite; absolute value no more than 1 trillion.
Common ratio: Required numeric input. finite; signed; absolute value no more than 1 million.
Target term number: Required numeric input. positive whole number no more than 1,000,000.

How to interpret the result

The calculator returns target term, finite series sum, growth factor, bounded term table, infinite sum or explicit non-convergence. Read each output against the assumptions you entered, compare alternatives one variable at a time, and use the result as an estimate rather than a promise.

Common mistakes to avoid

Mixing units or entering a rate as a decimal when the field requests a percentage.
Rounding intermediate values instead of the final displayed result.
Leaving out a cost, timing rule or local requirement listed in the assumptions below.

Practical uses

Use the geometric sequence calculator to check a real scenario, compare a changed assumption and explain the arithmetic before relying on a quote, specification or decision.

Worked example

For a₁=3, r=2 and n=5, a₅=3×2⁴=48 and S₅=3×(1−2⁵)/(1−2)=93. Because |2| is not below 1, the corresponding infinite series does not converge.

Assumptions and limitations

each term is obtained by multiplying the previous term by one constant ratio
the target index starts at one
all terms share one consistent unit
the infinite-series result assumes the sequence continues indefinitely

does not infer a sequence rule from a list of observations
does not solve ambiguous reverse-ratio roots
does not model arithmetic, recursive, random or irregular sequences
large ratios or term indices can exceed the supported finite numeric range
displayed decimals are rounded after full-precision calculation

Important notes

Results are estimates based on the values entered. Confirm rates, prices, units and professional requirements that apply in your location. Review the Answer Atlas Calculation Standard and corrections policy for formula, privacy and correction rules.

Frequently asked questions

What makes a sequence geometric?

Each nonzero term divided by the preceding term has the same constant value, called the common ratio.

When does an infinite geometric series converge?

A finite infinite-sum value exists only when the absolute value of the common ratio is less than 1.

Can the common ratio be negative or zero?

Yes. A negative ratio alternates signs, while a zero ratio makes every term after the first equal to zero.