How it works

This calculator helps you find the sum of a geometric series, whether it has a specific number of terms (finite) or goes on forever (infinite), as long as it converges. Just input the first term, common ratio, and optionally, the number of terms.

The Formula

For a finite geometric series with first term 'a', common ratio 'r', and 'n' terms:
S_n = a * (1 - r^n) / (1 - r) (where r ≠ 1) For an infinite geometric series:
S_∞ = a / (1 - r) (where |r| < 1)

Worked Example

1. Example: Sum of a Finite Geometric Series

   Let's find the sum of the first 5 terms of a geometric series where the first term (a) is 3 and the common ratio (r) is 2. Using the formula S_n = a * (1 - r^n) / (1 - r): S_5 = 3 * (1 - 2^5) / (1 - 2) S_5 = 3 * (1 - 32) / (-1) S_5 = 3 * (-31) / (-1) S_5 = 93 Our calculator will quickly give you this result.

Tips, Assumptions & Limitations

• For an infinite series, leave the 'Number of Terms' field blank.
• Remember, an infinite geometric series only has a finite sum if the absolute value of its common ratio (|r|) is less than 1.
• If the common ratio (r) is 1, the sum of a finite series is simply the first term multiplied by the number of terms (a * n).