Geometric Sequence Calculator

Last Updated on March 2, 2026 | 1 : 38 pm by Anas Brittany

Use this geometric sequence calculator to generate terms of a geometric sequence and find key values like the common ratio, the nth term, and the sum of the first n terms. Enter the first term, the common ratio, and how many terms you want to list.

What Is a Geometric Sequence?

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by the same constant value. That constant multiplier is called the common ratio.

Examples of geometric sequences:

• 2, 4, 8, 16, 32 … (ratio = 2)
• 81, 27, 9, 3, 1 … (ratio = 1/3)

Geometric sequences appear in finance, population growth, interest calculations, measurement scaling, and repeated percentage change.

Geometric Sequence Formula (Nth Term)

The nth term of a geometric sequence is:

aₙ = a₁ · r^(n−1)

Where:

• a₁ = first term
• r = common ratio
• n = term position (1st, 2nd, 3rd, etc.)

This formula lets you jump directly to any term without writing the entire sequence.

How to Find the Common Ratio

If you already have two terms, you can find the common ratio by dividing:

r = a₂ ÷ a₁

(As long as a₁ is not zero.)

If the ratio stays the same between each pair of consecutive terms, the sequence is geometric.

Sum of a Geometric Sequence

The sum of the first n terms is:

Sₙ = a₁ · (1 − r^n) / (1 − r) (when r ≠ 1)

If r = 1, the sequence is constant, so:

Sₙ = n · a₁

This is useful for calculating totals in repeated growth or repeated reduction problems.

Example

First term a₁ = 3, ratio r = 2:

Sequence:
3, 6, 12, 24, 48 …

5th term:
a₅ = 3 · 2^(5−1) = 3 · 16 = 48

Sum of first 5 terms:
3 + 6 + 12 + 24 + 48 = 93

Frequently Asked Questions

Can a geometric sequence have a negative ratio?

Yes. A negative ratio causes the terms to alternate between positive and negative values.

What if the ratio is a fraction?

That’s still a geometric sequence. The values shrink each term when |r| < 1.

What if the first term is zero?

If a₁ is 0, every term will be 0, and the ratio is not meaningful.