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JEE Main202126 Aug 2021Morning ShiftMathematicsSequences and SeriesActual

The sum of the series 1 x + 1 + 2 x 2 + 1 + 2 2 x 4 + 1 + … + 2 100 x 2 100 + 1 when x = 2 is:

Options

  1. A1 - 2 101 4 101 - 1
  2. B1 + 2 101 4 101 - 1
  3. C1 + 2 100 4 101 - 1
  4. D1 - 2 100 4 201 - 1

Correct answer

A. 1 - 2 101 4 101 - 1

Step-by-step solution

Adding and subtracting 1 x - 1 S = 1 x - 1 - 1 x - 1 + 1 x + 1 + 2 x 2 + 1 + 2 2 x 4 + 1 + … + 2 100 x 2 100 + 1 S = 1 x - 1 + 1 x + 1 - 1 x - 1 + 2 x 2 + 1 + 2 2 x 4 + 1 + … + 2 100 x 2 100 + 1 S = 1 x - 1 + 2 x 2 + 1 + - 2 x 2 - 1 + 2 2 x 4 + 1 + … . + 2 100 x 2 100 + 1 S = 1 x - 1 + - 2 2 x 4 - 1 + 2 2 x 4 + 1 + … . + 2 100 x 2 100 + 1 S = 1 x - 1 - 2 101 x 2 101 - 1 For x = 2 ⇒ S = 1 - 2 101 2 2 101 - 1 = 1 - 2 101 4 101 - 1

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