Can use Coordinate geometry

Consider a series of $n$ concentric circles ${C}_{1}, {C}_{2}, \ldots, {C}_{n}$ with radii ${r}_{1}, {r}_{2}, {r}_{3}, \ldots , {r}_{n}$ respectively satisfying ${r}_{1}>{r}_{2}>{r}_{3}> \cdots >{r}_{n}$ and
${r}_{1} = 10$ .

The circles are such that the chord of contact of tangents from any point on ${C}_{i}$ to ${C}_{i+1}$ is a tangent to ${C}_{i+2}$ where $i = 1, 2, 3, ...$ .

Find the value of $\displaystyle \ \lim_{ n\to \infty }{ \sum _{ r=1 }^{ n }{ { r }_{ i } } }$ , if the angle between the tangents from any point of ${C}_{1}$ to ${C}_{2}$ is $60^\circ$ .