Paragraph: Let M= \ (x, y) R R : x^ 2 +y^ 2 r^ 2 \ , where r0 . Consider the geometric progression a_ n = 1 2^ n-1 , n=1,2,3, . Let S_ 0 =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 ,

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Paragraph: Let M= \ (x, y) R R : x^ 2 +y^ 2 r^ 2 \ , where r>0 . Consider the geometric progression a_ n = 1 2^ n-1 , n=1,2,3, . Let S_ 0 =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= (2^ 199 -1 ) 2 2^ 198 . The number of all those circles D_ n that are inside M is

Quantrex Academy · Accepted Answer

Correct answer: B. 199

2 &#160; S n - 1 + a n &#60; 2 199 - 1 2 198 2 2 2 - 1 2 n - 2 + 1 2 n - 1 &#60; 2 199 - 1 2 198 2 2 2 - 1 2 n - 2 2 + 1 2 n - 1 &#60; 2 2 - 2 2 198 1 2 n - 2 1 2 - 2 &#60; - 2 2 198 1 2 n - 2 ( 2 2 - 1 ) 2 &#62; 2 2 198 2 n - 2 &#60; 2 - 1 2 2 197 n - 2 &#8804; 197 number of circle &#160; = 199

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