JEE Advanced2021MathematicsSequences and SeriesActual
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= (2¹⁹⁹-1 ) 2 2¹⁹⁸ . The num
Options
- A198
- B199
- C200
- D201
Correct answer
B. 199
Step-by-step solution
2   S n - 1 + a n < 2 199 - 1 2 198 2 2 2 - 1 2 n - 2 + 1 2 n - 1 < 2 199 - 1 2 198 2 2 2 - 1 2 n - 2 2 + 1 2 n - 1 < 2 2 - 2 2 198 1 2 n - 2 1 2 - 2 < - 2 2 198 1 2 n - 2 ( 2 2 - 1 ) 2 > 2 2 198 2 n - 2 < 2 - 1 2 2 197 n - 2 ≤ 197 number of circle   = 199