# Circles on Circle!

**Geometry**Level 5

We define a sequence of circles \(C_0, C_1, C_2, \ldots, C_n\) in the Cartesian plane as follows:

\(C_0\) is the circle \(x^2+y^2=1\).

for \(n = 0,1,2 \ldots\), the circle \(C_{n+1}\) lies in upper half plane and is tangent to \(C_n\) as well as to both branches of the hyperbola \(x^2-y^2=1\).

Let \(r_n\) denote the radius of the circle \(C_n\)

Then evaluate \[\lim_{n \to \infty} \frac{r_n}{r_{n-1}} .\]

Round off your answer to 2 decimal places.