Geometric Sequence Calculator
Calculate any term and the sum of the first n terms of a geometric sequence — a sequence where each term is multiplied by the same fixed ratio to get the next one.
- Free to use
- Accurate results
- No registration required
- Works on all devices
Enter the first term, common ratio, and term number to calculate the nth term and the sum of the first n terms of a geometric sequence.
The fixed factor each term is multiplied by to get the next one.
Which term to find, and how many terms to include in the sum.
Your result
162
nth term
Sum of first n terms242
AI explanation
Formula
nth term = a₁ × r^(n−1); Sum = a₁ × (rⁿ − 1) / (r − 1), or a₁ × n when r = 1Worked example
2, 6, 18, 54, ... find the 5th term
| Field | Value |
|---|---|
| First term (a₁) | 2 |
| Common ratio (r) | 3 |
| Term number (n) | 5 |
| nth term | 162 |
| Sum of first n terms | 242 |
Assumptions
- The common ratio is applied consistently between every consecutive pair of terms.
- The first term cannot be zero.
- Very large term numbers combined with a large ratio can grow beyond what can be computed precisely — this calculator will show an error in that case rather than an inaccurate result.
Frequently asked questions
What is a geometric sequence?
It's a sequence of numbers where each term is found by multiplying the previous one by a fixed value, called the "common ratio" — for example, 2, 6, 18, 54 has a common ratio of 3.
How is the sum of a geometric sequence calculated?
Sum = a₁ × (rⁿ − 1) / (r − 1) for any ratio other than 1. When the ratio is exactly 1, every term equals the first term, so the sum is simply a₁ × n.
What does a negative common ratio do?
It makes the sequence alternate in sign — for example, with a first term of 1 and a ratio of −2, the sequence is 1, −2, 4, −8, 16, ...