Geometric Sequence Calculator
Calculate nth term and sum of geometric sequences
How to Use This Calculator
Select What to Find
Choose whether you want to find the nth term or the sum of the first n terms.
Enter the Values
Input the first term (a₁), common ratio (r), and term number (n). The common ratio cannot be 0, and n must be a positive integer.
Click Calculate
Press the "Calculate" button to see the result displayed below.
Formula
nth Term: a_n = a₁ × rⁿ⁻¹
Sum: S_n = a₁ × (1 - rⁿ) / (1 - r), for r ≠ 1
If r = 1, then S_n = a₁ × n
Where:
- a_n = nth term
- a₁ = first term
- r = common ratio
- n = term number
- S_n = sum of first n terms
Example: Find the 5th term of 2, 6, 18, 54, ...
a₁ = 2, r = 3, n = 5
a₅ = 2 × 3⁴ = 2 × 81 = 162
About Geometric Sequence Calculator
A geometric sequence (also called geometric progression) is a sequence where each term after the first is obtained by multiplying the previous term by a constant value called the common ratio. The geometric sequence calculator helps you find any term in the sequence and calculate the sum of the first n terms.
Key Features
- Find nth Term: Calculate any term in a geometric sequence
- Calculate Sum: Find the sum of the first n terms
- Simple Input: Just enter first term, common ratio, and term number
- Instant Results: Get accurate calculations immediately
Common Applications
- Financial calculations (compound interest, investments)
- Population growth and decay models
- Mathematical sequences and series problems
- Exponential growth and decay patterns
Frequently Asked Questions
What is a geometric sequence?
A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a constant (common ratio). Example: 2, 6, 18, 54, 162, ... where the common ratio is 3.
Can the common ratio be negative?
Yes! A negative common ratio creates an alternating sequence. Example: 2, -6, 18, -54, ... where r = -3.
What if the common ratio is 1?
If r = 1, all terms are the same (constant sequence), and the sum is simply S_n = a₁ × n.
Can the common ratio be between 0 and 1?
Yes! If 0 < r < 1, the sequence decreases. Example: 100, 50, 25, 12.5, ... where r = 0.5.
How do I find the infinite sum?
For infinite geometric series, if |r| < 1, the sum converges to S_∞ = a₁ / (1 - r). This calculator focuses on finite sums.