Geometric Sequence Calculator

Explore terms, finite sums and convergence with an exact geometric sequence and a live term plot.

Calculation inputs

Precision & interpretation

Decimal inputs are stored as exact fractions. The displayed chart uses floating-point coordinates and at most the first 30 terms; the result uses all n terms.

Calculation results

Nth term4374a8 · exact integer

Finite sum Sₙ6560

Infinite sumDoes not converge

Terms included8

Step-by-step solution

Choose the formula
$aₙ = a₁rⁿ⁻¹$
Substitute your inputs
$a8 = (2) × (3)^7$
Calculate in order
$Multiply by the common ratio 7 times.$
Write the result
$a8 = 4374$

Sequence explorer

Discrete terms · linear axes · first 8 terms shown. The result and sum include all 8 terms.

Finite sum: 6560. A nonzero geometric series needs |r| < 1 to converge.

Exact value & calculation details

4374

Quantity	Value
Nth term	4374
Finite sum	6560
Infinite sum	Does not converge
Term 1	2
Term 2	6
Term 3	18
Term 4	54
Term 5	162
Term 6	486
Term 7	1458
Term 8	4374

Understand the calculation

Concept & meaning

Each term is obtained by multiplying the previous term by the same ratio. The nth term and the sum are different quantities.

Formula & special cases

For r = 1 the finite sum is n × a₁. For |r| < 1 the infinite series converges to a₁/(1 − r); otherwise it generally diverges. The all-zero sequence also sums to zero.

Precision & limitations

Decimal inputs are stored as exact fractions. The displayed chart uses floating-point coordinates and at most the first 30 terms; the result uses all n terms.

a_{n} = a_{1} r^{n - 1}

Worked example from your inputs

a8 · exact integer = 4374

Frequently asked questions

Can the common ratio be negative?

Yes. A negative ratio alternates the sign of consecutive nonzero terms.

What happens when r equals 1?

Every term equals a₁ and the sum is n × a₁. The formula with division by 1 − r is replaced by this case.

When does the infinite sum exist?

For a nonzero first term, it exists only when |r| < 1. If a₁ is zero, every term is zero for any finite ratio.

Are the values exact?

Inputs with up to six decimal places are treated as rational numbers. A noninteger result may show a rounded decimal alongside its exact fraction.

Show all questions (5 more)Show fewer questions

What does the Geometric Sequence calculate?

Explore terms, finite sums and convergence with an exact geometric sequence and a live term plot.

Which values should I enter?

Use the labeled fields above. Decimals: up to 6 places. Values outside the supported range clear the result and show an explanation.

What formula does this calculator use?

aₙ = a₁rⁿ⁻¹; Sₙ = a₁(1 − rⁿ)/(1 − r)

Why might my result differ from another source?

Results can differ when another source uses different units, rounding rules, conventions or assumptions. Compare the inputs and definitions before comparing results.

Does the result update automatically?

Yes. Inputs and presets update the result, teaching diagram and expandable steps together. Invalid input clears the previous result.

Sources & References

$OpenStax official logo$ OpenStax · Geometric sequencesFinite and convergent infinite geometric seriesRead the reference

References explain the mathematics; they do not imply institutional endorsement.