MATH · GEOMETRIC SEQUENCE
Geometric Sequence Calculator
Find the Nth term and sum of a geometric or arithmetic sequence. Enter the first term, common ratio (or difference), and N to get results instantly.
Term 5
162
Sum S(5) = 242
|r| > 1 — sequence diverges
First term (a₁)2
Ratio (r)3
Term 5162
Sum S(5)242
2, 6, 18, 54, 162
About This Calculator
Compute the Nth term and the sum of the first N terms for a geometric or arithmetic sequence. Enter your first term, common ratio (or difference), and N to instantly see the result and the first 20 terms listed out.
How It Works
For a geometric sequence with first term a₁ and ratio r, the Nth term is a₁ × r^(n−1) and the sum is a₁ × (1 − rⁿ) / (1 − r) for r ≠ 1. For an arithmetic sequence with first term a₁ and common difference d, the Nth term is a₁ + (n−1)d and the sum is n/2 × (2a₁ + (n−1)d).
The Formula
a(n) = a₁ × r^(n-1)
- a₁
- first term
- r
- common ratio (geometric) or d for difference (arithmetic)
- n
- term number
Frequently Asked Questions
- What is a geometric sequence?
- A geometric sequence is a sequence where each term is multiplied by a fixed number (the common ratio r) to get the next term. For example, 2, 6, 18, 54, ... has a common ratio of 3.
- What is the difference between a geometric and arithmetic sequence?
- In a geometric sequence, consecutive terms have a constant ratio (multiplication). In an arithmetic sequence, consecutive terms have a constant difference (addition). For example, 2, 4, 8, 16 is geometric (×2); 2, 4, 6, 8 is arithmetic (+2).
- What happens when the ratio is greater than 1?
- When |r| > 1, the terms grow without bound — the sequence diverges. The sum S(n) grows rapidly. The calculator shows a notice when |r| > 1 so you are aware the series does not converge to a finite limit.
- Can I use negative or decimal ratios?
- Yes. A negative ratio alternates the sign of each term (e.g. r = −2 gives 1, −2, 4, −8, ...). A decimal ratio between 0 and 1 produces a converging sequence whose sum approaches a finite limit (geometric series).