Can use Coordinate geometry

Geometry Level 4

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 \(60^\circ\).


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