Geometric Sequence Calculator
Find any term, the sum of n terms, or the infinite sum of a geometric sequence. Enter the first term and common ratio to see the formula and result.
Geometric Sequence
Compute a single term a_n from a_1, r, and n.
Formula
Nth term (a_n)
162
First term (a_1)
2
Common ratio (r)
3
Term count (n)
5
First 5 terms
2, 6, 18, 54, 162
Frequently Asked Questions about the Geometric Sequence Calculator
What is a geometric sequence?
A geometric sequence multiplies each term by the same fixed number, called the common ratio r. The nth term is a_n = a_1 * r^(n - 1). For example, 2, 6, 18, 54 has a_1 = 2 and r = 3.
When does an infinite geometric series converge?
It converges only when |r| < 1. The closed form is S = a_1 / (1 - r). If |r| is 1 or larger, the partial sums grow without bound and there is no finite total.
What is the formula for the sum of the first n terms?
Use S_n = a_1 * (1 - r^n) / (1 - r) when r is not 1. If r equals 1, every term is a_1, so S_n is just n * a_1. The calculator handles both cases for you.
Can the common ratio be negative or zero?
Yes. A negative r produces an alternating sign pattern, like 1, -2, 4, -8. An r of 0 gives a_1, 0, 0, 0 and after the first term every value is zero. Both are valid inputs.
How is this different from an arithmetic sequence?
An arithmetic sequence adds the same constant d each step (linear growth). A geometric sequence multiplies by the same ratio r each step (exponential growth). They share the index n but use different closed-form formulas.