Geometric Sequence Generator
Generate a geometric sequence online by entering its first term and common ratio. Calculate individual terms, the sum of the first n terms, and the infinite sum when the series converges.
Generate your sequence
Enter the first term, common ratio, and number of terms. You can use positive numbers, negative numbers, decimals, or zero.
What is a geometric sequence?
A geometric sequence is a sequence in which every term after the first is found by multiplying the previous term by a fixed number called the common ratio.
For example, 2, 6, 18, 54, 162, ... is geometric because each term is three times the preceding term. The first term is 2 and the common ratio is 3.
Geometric sequence formulas
Find the nth term
The formula calculates a particular term directly. For example, if a₁ = 2 and r = 3, then the fifth term is 2 × 3⁴ = 162.
Sum of the first n terms
When r = 1, all terms are identical and Sₙ = n × a₁.
Infinite geometric series
An infinite geometric series has a finite sum when the absolute value of its common ratio is less than 1. Otherwise, the series does not converge to a finite sum, except that an all-zero sequence sums to zero.
Examples of geometric sequences
Geometric growth
The common ratio is 3, so the magnitude increases.
Geometric decay
The common ratio is 0.5, so the terms approach zero.
Alternating signs
The common ratio is -2, so the signs alternate.
Constant sequence
The common ratio is 1.
Frequently asked questions
What is a geometric sequence?
A geometric sequence is a sequence in which each term is obtained by multiplying the preceding term by a constant called the common ratio.
What is the formula for the nth term?
The nth term is a_n = a_1 times r raised to the power n minus 1, where a_1 is the first term and r is the common ratio.
How is the sum of a geometric sequence calculated?
For r not equal to 1, S_n = a_1(1-r^n)/(1-r). When r equals 1, the sum is n times a_1.
Can a geometric sequence have a negative ratio?
Yes. A negative common ratio makes the signs alternate when the first term is nonzero.
When does an infinite geometric series converge?
An infinite geometric series converges when the absolute value of the common ratio is less than 1. Its sum is a_1 divided by 1-r.