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JEE Main202118 Mar 2021Morning ShiftMathematicsSequences and SeriesActual

1 3 2 - 1 + 1 5 2 - 1 + 1 7 2 - 1 + … + 1 ( 201 ) 2 - 1 is equal to

Options

  1. A101 404
  2. B25 101
  3. C101 408
  4. D99 400

Correct answer

B. 25 101

Step-by-step solution

Given S = 1 3 2 - 1 + 1 5 2 - 1 + 1 7 2 - 1 + … + 1 ( 201 ) 2 - 1 T n = 1 ( 2 n + 1 ) 2 - 1 = 1 4 ( n ) ( n + 1 ) T n = 1 4 ( n + 1 ) - n n ( n + 1 ) = 1 4 1 n - 1 n + 1 ∴   S = ∑ n = 1 100 T n = ∑ n = 1 100 1 4 1 n - 1 n + 1 ∴   S = 1 4 1 - 1 2 + 1 2 - 1 3 + . . . + 1 100 - 1 101 ⇒ S = 1 4 1 - 1 101 = 1 4 100 101 = 25 101

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