Question 121
Two rays have their common origin at the point $O$. A circumference $C_{1}$ is considered tangent to both half-straights, whose center is located at a distance $d_{1}$ of $O$, and whose radius is $r_{1}$. The circumferences are built successively $C_{n}$, so that $C_{n}$ is tangent to the half-straights, outer tangent to $C_{n-1}$ and such that the distance from its center to $O, d_{n}$, is less than $d_{n-1}$, for $n\gt 1$. Find the sum of the areas of the circles limited by the circumferences $C_{n}$, for all $n$, depending on $r_{1}$ and $d_{1}$.