Geometric Progression Calculator
Calculate the n-th term, partial sum, and infinite sum of a geometric progression. Enter the first term, common ratio, and term number.
aₙ = a₁ · r^(n−1)
How to Use the Geometric Progression Calculator
- Enter the numbers or values in the input fields.
- The result is calculated and displayed automatically.
- Review the step-by-step solution or detailed breakdown.
- Copy the result or adjust inputs for a new calculation.
Quick Reference
| From | To |
|---|---|
| 2 + 3 | 5 |
| 12 × 12 | 144 |
| √144 | 12 |
| 2¹⁰ | 1,024 |
| π | 3.14159 |
| e | 2.71828 |
Use Cases
- •Checking homework or exam answers quickly and accurately.
- •Verifying manual calculations in professional or academic work.
- •Learning mathematical concepts with instant visual feedback.
- •Performing quick computations during meetings or presentations.
Formula
A geometric progression has a constant ratio r between consecutive terms. n-th term: aₙ = a₁·r^(n−1). Sum of n terms: Sₙ = a₁·(rⁿ − 1)/(r − 1) for r ≠ 1. Infinite sum (|r| < 1): S∞ = a₁/(1 − r).
Frequently Asked Questions
What is a geometric progression?
A geometric progression (GP) is a sequence where each term is obtained by multiplying the previous term by a fixed ratio r. Example: 3, 6, 12, 24, ... with r = 2.
When does the infinite sum converge?
The infinite sum of a geometric series converges only when the absolute value of the common ratio is less than 1 (|r| < 1). The sum is S∞ = a₁/(1 − r).
What happens when r = 1?
When r = 1, every term equals a₁, and the sum of n terms is simply n·a₁. The standard sum formula is not used because division by zero would occur.
Can the common ratio be negative?
Yes. A negative ratio means the terms alternate in sign. For example: 1, −2, 4, −8, ... with r = −2.