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Geometric Sequence Calculator

Find the nth term and sum of a geometric sequence from the first term, common ratio and n. Includes the infinite sum when the ratio is between −1 and 1.

486
Term 6 (aₙ)
728
Sum of first 6 terms
diverges
Infinite sum

Sequence & formulas

2, 6, 18, 54, 162, 486

aₙ = a₁ · r^(n − 1) = 2 · 3^(6 − 1) = 486

The infinite sum only converges when |r| < 1.

What is the Geometric Sequence Calculator?

The ByteTools Geometric Sequence Calculator handles progressions where each term is multiplied by a constant ratio.

  • nth term and sum of the first n terms
  • Infinite sum shown when the series converges
  • Lists the opening terms of the sequence
  • Shows the geometric formulas used
  • Handles negative and fractional ratios
  • 100% private — runs entirely in your browser

How to use the Geometric Sequence Calculator

  1. 1

    Enter the first term of the sequence.

  2. 2

    Enter the common ratio between consecutive terms.

  3. 3

    Enter n, the number of terms.

  4. 4

    Read the nth term, the finite sum and the listed terms.

  5. 5

    When the ratio is between −1 and 1, note the infinite sum too, then click Copy.

About the Geometric Sequence Calculator

The ByteTools Geometric Sequence Calculator handles progressions where each term is multiplied by a constant ratio. Enter the first term, the common ratio and n, and it returns the nth term, the sum of the first n terms and the opening terms, plus the infinite sum when the series converges.

It is useful for students, financial modelling of compound growth and anyone dealing with exponential patterns. When the ratio lies between −1 and 1 the tool also reports the sum to infinity, which is where geometric series get interesting.

All the maths runs in your browser with JavaScript. Nothing is uploaded, so it is private, instant and works offline. Copy the results with one click.

Frequently asked questions

What is a geometric sequence?

A geometric sequence is a list where each term is the previous one multiplied by a fixed number called the common ratio. For example 2, 6, 18, 54 has a common ratio of 3.

How do you find the nth term of a geometric sequence?

Use aₙ = a₁ × r^(n − 1), where a₁ is the first term and r is the common ratio. For 2, 6, 18 the 5th term is 2 × 3⁴ = 162. The calculator computes this power for you.

How do you find the sum of a geometric sequence?

For r not equal to 1, the sum of the first n terms is Sₙ = a₁ × (1 − rⁿ) ÷ (1 − r). If r equals 1 every term is the same, so the sum is simply n times the first term. The tool handles both cases.

When does a geometric series have an infinite sum?

When the absolute value of the common ratio is less than 1, the terms shrink toward zero and the infinite sum converges to a₁ ÷ (1 − r). If the ratio is 1 or larger in magnitude, the sum grows without bound.

Can the common ratio be negative?

Yes. A negative ratio makes the terms alternate in sign, like 3, −6, 12, −24. The calculator handles negative and fractional ratios and still reports the infinite sum when the magnitude is below 1.

Guides for Geometric Sequence Calculator

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