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JEE Main202229 Jul 2022Evening ShiftMathematicsSequences and SeriesActual

∑ r = 1 20 r 2 + 1 r ! is equal to

Options

  1. A22 ! - 21 !
  2. B22 ! - 2 21 !
  3. C21 ! - 2 20 !
  4. D21 ! - 20 !

Correct answer

B. 22 ! - 2 21 !

Step-by-step solution

Given, ∑ x = 1 20 r 2 + 1 r ! Now on rearranging we get, = ∑ x = 1 20 ( r + 1 ) 2 - 2 r r ! = ∑ x = 1 20 r + 1 r + 1 ! - r · r ! - ∑ r = 1 20 r · r ! = ∑ x = 1 20 r + 1 r + 1 ! - r . r ! - ∑ r = 1 20 r + 1 ! - r ! = 21 . 21 - 1 - 21 - 1 = 21 . 21 - 21 = 20 . 21 ! = 22 - 2 21 ! = 22 ! - 2 . 21 !

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