Free Online Geometric Series Calculator

How to Use

1. Enter First Term (a): The initial value of the sequence
2. Enter Common Ratio (r): The multiplier between consecutive terms
3. Enter Number of Terms (n): How many terms to sum
4. Click Calculate: Get the sum and detailed explanation

Understanding Geometric Series

A geometric series is the sum of terms in a geometric sequence, where each term is found by multiplying the previous term by a constant called the common ratio.

Formula

Finite Series: S_n = a(1 - rⁿ) / (1 - r) when r ≠ 1

When r = 1: S_n = n × a

Infinite Series (|r| < 1): S = a / (1 - r)

Example

Sum of 2 + 6 + 18 + 54 + 162:

• First term (a) = 2
• Common ratio (r) = 3
• Number of terms (n) = 5
• Sum = 2(1 - 3⁵) / (1 - 3) = 2(-242) / (-2) = 242

Key Features

• Calculate finite geometric series sums
• Evaluate infinite series convergence
• Step-by-step solution display
• Support for negative ratios
• Handles fractional ratios
• Shows all terms in the sequence