Mathematics - Sequence & Series Question with Solution | TestHub

Paragraph: Let $M=\left\{(x,y)\in \mathbb{R}\times \mathbb{R}:x^2+y^2\le r^2\right\}$, where $r>0.$ Consider the geometric progression $a_n=\frac{1}{2^{n-1}},n=1,2,3,\dots .$ Let $S_0=0$ and, for $n\ge 1$, let $S_n$ denote the sum of the first $n$ terms of this progression. For $n\ge 1$, let $C_n$ denote the circle with center $\left(S_{n-1},0\right)$ and radius $a_n$, and $D_n$ denote the circle with center $\left(S_{n-1},S_{n-1}\right)$ and radius $a_n$.
Question: Consider $M$ with $r=\frac{1025}{513}$. Let $k$ be the number of all those circles $C_n$ that are inside $M$. Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then