# 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.

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