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

Use the formula an = a1 multiplied by r raised to the power n minus 1.

Free Sequence Calculation Tool

Geometric Sequence Calculator

Calculate the nth term, finite sum, infinite sum, convergence status and generated values of any geometric sequence.

Instant calculation Infinite sum test Mobile friendly

Geometric Sequence Calculator

Enter the first term, common ratio and term position

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Nth term aₙ = a₁rⁿ⁻¹

Finite sum Sₙ = a₁(1 − rⁿ)/(1 − r)

Infinite sum S∞ = a₁/(1 − r), |r| < 1

First Term a₁

The starting value of the sequence.

Common Ratio r

The constant multiplier between terms.

Term Number n

The position and number of terms to calculate.

Nth Term

Value of the requested term.

Finite Sum

Sum of the first requested terms.

Infinite Sum

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Available only when |r| is less than 1.

Generated Sequence

Your generated sequence will appear here.

Nth term calculation aₙ = a₁ × r^(n − 1)

Finite sum calculation Sₙ = a₁(1 − rⁿ)/(1 − r)

Try an example

What Is a Geometric Sequence Calculator?

A geometric sequence calculator finds values in a number pattern where each term is produced by multiplying the previous term by the same constant value. This constant multiplier is known as the common ratio.

Enter the first term, common ratio and required term position to calculate the nth term, finite series sum, infinite sum and a preview of generated terms.

The tool supports increasing, decreasing, alternating, fractional and constant geometric sequences.

Nth term calculation Find the value at any requested position in the sequence.

Finite series sum Calculate the total of the first n geometric terms.

Convergence checking Determine whether an infinite geometric series has a finite sum.

Geometric Sequence Formulas

These formulas calculate individual terms, finite sums and convergent infinite sums.

Nth term aₙ = a₁rⁿ⁻¹

Finite sum Sₙ = a₁(1 − rⁿ)/(1 − r)

Infinite sum S∞ = a₁/(1 − r)

Common ratio r = aₙ/aₙ₋₁

Simple Process

How the Geometric Sequence Calculator Works

Calculate geometric sequence terms and sums in three simple steps.

Enter the First Term

Add the starting number of the geometric sequence. This value is represented by a₁.

Add the Common Ratio

Enter the fixed number used to multiply each term to produce the next term.

Choose the Term Number

Enter the required position to calculate the nth term, sums and sequence preview.

Formula Guide

Geometric Sequence Formulas Explained

Use these formulas to calculate individual terms and the total value of geometric series.

Nth Term

The nth term formula finds the value at a particular position.

aₙ = a₁rⁿ⁻¹

aₙ Value of the required term
a₁ First term of the sequence
r Common ratio
n Term position

Finite Sum

The finite sum formula adds the first n terms of the sequence.

Sₙ = a₁(1 − rⁿ)/(1 − r)

Sₙ Sum of the first n terms
a₁ First term
r Common ratio
n Number of terms being added

Infinite Sum

An infinite sum exists only when the absolute value of the ratio is less than one.

S∞ = a₁/(1 − r)

S∞ Sum of infinitely many terms
|r| Must be less than 1
r May be positive or negative
a₁ Starting term

Worked Examples

Geometric Sequence Examples

Compare increasing, decreasing, alternating and convergent geometric sequences.

First Term	Ratio	Term Number	Nth Term	Finite Sum	Sequence Preview
3	2	8	384	765	3, 6, 12, 24, 48...
100	0.5	5	6.25	193.75	100, 50, 25, 12.5, 6.25
4	−2	5	64	44	4, −8, 16, −32, 64
5	1	6	5	30	5, 5, 5, 5, 5, 5
8	0.25	4	0.125	10.625	8, 2, 0.5, 0.125

Helpful Answers

Geometric Sequence Calculator FAQs

Find answers to common questions about geometric terms, ratios and series.

What is a geometric sequence?

A geometric sequence is a number pattern in which each term is found by multiplying the previous term by the same common ratio.

How do I find the nth term?

Use aₙ = a₁rⁿ⁻¹. Raise the common ratio to one less than the required term position and multiply the result by the first term.

How do I calculate the common ratio?

Divide any term by the term immediately before it. For example, in 3, 12, 48, the common ratio is 12 ÷ 3 = 4.

Can the common ratio be negative?

Yes. A negative ratio creates an alternating sequence where the signs change between positive and negative values.

When does an infinite geometric series converge?

An infinite geometric series converges only when the absolute value of the common ratio is less than 1. For example, ratios of 0.5 and −0.25 converge.

What happens when the common ratio is 1?

Every sequence term remains equal to the first term. The finite sum is therefore the first term multiplied by the number of terms.

What is the difference between a sequence and a series?

A sequence is the ordered list of terms. A series is the result of adding those terms together. This tool calculates both.

Calculate Any Geometric Sequence Instantly

Enter your sequence values above to calculate the nth term, finite sum, infinite sum and generated terms.

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