Math
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 follows a_n = a_1 x r^(n - 1). For example, 2, 6, 18, 54 has a first term a_1 = 2 and r = 3, because each term is three times the one before it.
When does an infinite geometric series converge?
It converges only when |r| is strictly less than 1 - for example, r = 0.5 or r = -0.25. The sum equals a_1 / (1 - r). When |r| is 1 or greater, the terms never shrink fast enough and the series has no finite total, so the calculator returns no result.
What is the formula for the sum of the first n terms?
When r is not equal to 1, the sum is S_n = a_1 x (1 - r^n) / (1 - r). When r equals exactly 1, every term is a_1, so the sum simplifies to n x a_1. The calculator applies whichever case fits your inputs automatically.
Can the common ratio be negative or zero?
Yes. A negative r produces alternating signs - for example, a_1 = 1 and r = -2 gives 1, -2, 4, -8. A ratio of exactly 0 yields the first term followed by all zeros. Both are valid inputs and the calculator handles them correctly.
How is this different from an arithmetic sequence?
An arithmetic sequence adds the same constant d each step, producing linear growth (1, 4, 7, 10 with d = 3). A geometric sequence multiplies by r each step, producing exponential growth or decay. The two share the same index n but use entirely different formulas for both the nth term and the partial sum.