What is the formula for the sum to infinite terms of a geometric progression whose first term is a and common ratio r?

The formula for the sum to infinity (S_∞) of a geometric progression (GP) with the first term 'a' and common ratio 'r' is:

S_∞ = a / (1 - r)

Understanding the Formula

This formula is valid only when the absolute value of the common ratio 'r' is less than 1 (i.e., |r| < 1). This condition ensures that the terms of the geometric progression become progressively smaller, and the sum converges to a finite value. If |r| ≥ 1, the sum to infinity does not exist, as the series either diverges to infinity or oscillates.

Derivation (Brief Explanation)

The formula is derived from the formula for the sum of the first 'n' terms of a GP:

S_n = a(1 - rⁿ) / (1 - r)

As n approaches infinity, if |r| < 1, then rⁿ approaches 0. Therefore:

S_∞ = lim (n→∞) S_n = a(1 - 0) / (1 - r) = a / (1 - r)

Condition for Convergence

• |r| < 1: The series converges, and the sum to infinity exists.
• |r| ≥ 1: The series diverges, and the sum to infinity does not exist.

Example

Let's consider a GP where the first term (a) is 4 and the common ratio (r) is 1/2. Since |1/2| < 1, the sum to infinity exists.

S_∞ = 4 / (1 - 1/2) = 4 / (1/2) = 8

Therefore, the sum to infinity of this GP is 8.