Paragraph: Let M= (x, y) R R : x²+y² r² , where r>0 . Consider the geometric progression a_ n = 1 2^ n-1 , n=1,2,3, . Let S₀=0 and, for n 1 , let S_ n denote the sum of the first n terms of this progression. For n 1 , let C_ n denote the circle with center (S_ n-1 , 0 ) and radius a_ n , and D_ n denote the circle with center (S_ n-1 , S_ n-1 ) and radius a_ n . Question: Consider M with r= 1025 513 . Let k be the nu

Step-by-step solution

S n = 1 + 1 2 + 1 2 2 + … + 1 2 n - 1 = 2 1 - 1 2 n = 2 - 1 2 n - 1 Centre of C n is 2 - 1 2 n - 2 , 0 and radius of C n is 1 2 n - 1 when r = 1025   513 < 2 C n will lie inside m when 2 - 1 2 n - 2 + 1 2 n - 1 < 1025   513 ⇒ 1 - 1 2 n < 1025 1026 ⇒ 2 n < 1026 ⇒ n ≤ 10 Hence number of circles ⇒ k = 10 Also ℓ = 5 3 k + 2 ℓ = 30 + 10 = 40