Algebra
Intermediate

Geometric Series Sum

Sum of the first n terms of a geometric sequence (r ≠ 1).

Formula

Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}

Variables

a_1First term
rCommon ratio
nNumber of terms

Example

Sum of 2 + 6 + 18 + 54 (a₁=2, r=3, n=4): S = 2·(1−81)/(1−3) = 80.

Did You Know?

The chessboard-and-rice legend — doubling grains on each square — needs more rice than exists on Earth, thanks to geometric growth.

geometricseriessequence