Geometric series

For first term a and ratio r, the first n terms sum to a(1 − rⁿ)/(1 − r) when r ≠ 1. If r = 1, every term equals a and the sum is na.

An infinite geometric series converges only when |r| < 1; its sum is a/(1 − r). A negative ratio alternates signs. Convergence is a condition to check before substituting into the infinite-sum formula.

Worked example

Find 12 + 6 + 3 + … to infinity.

1. The first term is 12 and the ratio is 1/2.
2. Since |1/2| < 1, the series converges.
3. Use 12/(1 − 1/2).

Answer: 24