Geometric Series Sum Calculator

About this calculator

The sum of a geometric series — the first n terms of a geometric sequence added together — has a closed-form shortcut that avoids adding every term one by one: S_n = a₁(1 − r^n) / (1 − r) for a common ratio r not equal to 1, where a₁ is the first term. This formula comes from a classic algebraic trick of subtracting the series from itself shifted by one factor of r, which cancels every term except the first and last.

The formula underlies the present-value calculation for a fixed-payment annuity or loan in finance (each payment is a term in a geometric series discounted by a constant factor each period), and it's a standard topic in algebra and precalculus courses as the natural extension of finding a single geometric-sequence term to summing a whole run of them.

This calculator takes the first term, common ratio, and number of terms and returns the sum of the series.