jmc

Question

Let x and y be real numbers such that -1 < x < y < 1. Let G be the sum of the infinite geometric series whose first term is x and whose common ratio is y, and let G' be the sum of the infinite geometric series whose first term is y and whose common ratio is x. If G = G', find x + y.

Accepted Answer

We have that G = x1 - y and G' = y1 - x, so x1 - y = y1 - x.Then x(1 - x) = y(1 - y), so x - x^2 = y - y^2. Then x^2 - y^2 + y - x = 0. We can factor this as (x - y)(x + y) - (x - y) = 0,so (x - y)(x + y - 1) = 0. Since x < y, we must have x + y = 1.