Using Descartes' circle theorem, it's easier to work with the curvature rather than the radius. Let . Then the Descartes theorem tells you that (see equation (2) in the Wikipedia article). You want the circle with the smaller radius, therefore the larger curvature, so take the positive square root: .

It's now easy to prove by induction that . You get the base case k_3 = 4/r by taking in the formula . The inductive step goes like this.