Paragraph: Let M=\ (x, y) ∈ ℝ × ℝ: x^ 2 +y^ 2 ≤ r^ 2 \ , where r>0...

{S}_{n} = 1 + \dfrac{1}{2} + \dfrac{1}{{2}^{2}} + \ldots + \dfrac{1}{{2}^{n - 1}}$</p><p>$= 2\left(1 - \dfrac{1}{{2}^{n}}\right) = 2 - \dfrac{1}{{2}^{n - 1}}$</p><p>Centre of${C}_{n}$is \left(2 - \dfrac{1}{{2}^{n - 2}}, 0\right)$

and radius of ${C}_{n}$ is $\dfrac{1}{{2}^{n - 1}}$

when $r = \dfrac{1025}{513} < 2$

${C}_{n}$ will lie inside $m$

when $2 - \dfrac{1}{{2}^{n - 2}} + \dfrac{1}{{2}^{n - 1}} < \dfrac{1025}{513}$

$⇒ 1 - {\left(\dfrac{1}{2}\right)}^{n} < \dfrac{1025}{1026}$

$⇒ {2}^{n} < 1026 ⇒ n \le 10$

Hence number of circles

$⇒ k = 10$

Also $ℓ = 5$

$3k + 2ℓ = 30 + 10 = 40$