What is the common ratio?

It is the fixed multiplier from one term to the next. For two known first terms, the ratio is the second term divided by the first term.

When does the infinite geometric series have a sum?

The infinite sum converges only when the absolute value of the common ratio is less than 1.

Can the common ratio be negative?

Yes. A negative common ratio creates alternating signs in the sequence.

How is a geometric sequence different from an arithmetic sequence?

A geometric sequence multiplies by a fixed ratio; an arithmetic sequence adds a fixed difference.

Geometric Sequence Calculator

Calculate a geometric sequence from the first term and common ratio, or from the first two terms. See the nth term, first n-term sum, and infinite sum when it converges.

What do you know?

First term b1

Common ratio r

Number of terms to show (N)

Geometric sequence means each term is found by multiplying the previous term by the same common ratio. The calculator also reports the infinite series sum when |r| is less than 1.

What this geometric sequence calculator finds

Search results for geometric sequence calculators focus on the common ratio, nth term, partial sum, infinite series sum, and a preview of the generated terms.

Enter first term and common ratio when the sequence rule is known.
Enter the first two terms when the common ratio should be calculated.
Check the infinite sum only when the absolute value of the common ratio is less than 1.

Geometric sequence formulas

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

a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the term number.

S_{n} = \frac{a _{1} ( r ^{n} - 1 )}{r - 1}

S_n is the sum of the first n terms when r is not 1.

S_{n} = n a_{1}

Use this version when the common ratio r equals 1.

S_{\infty} = \frac{a _{1}}{1 - r}

S_infinity exists only when the absolute value of r is less than 1.

Common cases

Common ratio	Meaning
	Terms grow in size if the first term is positive
	Terms shrink toward zero
	Every term is the same
	All terms after the first are zero
	Signs alternate

If the first term is zero, the ratio cannot be inferred from the first two terms because division by zero would be required.

Frequently Asked Questions

Sources and References

Calculations are based on the listed reference sources. Links open in a new tab.

Updated: June 2, 2026

Related Tools

Arithmetic Sequence Calculator

Calculate an arithmetic sequence from the first term and common difference, or from the first two terms. See the nth term, the sum of the first terms, and the generated sequence.

Open

Arithmetic Sequence Nth Term Calculator

Find the nth term of an arithmetic sequence from the first term, common difference, and term number. The calculator also shows the substitution and a short sequence check.

Open

Arithmetic Series Sum Calculator

Calculate the sum of the first n terms of an arithmetic series from the first term and common difference, or from the first and last terms.

Open