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JEE Main2016MathematicsBinomial TheoremActual

The sum ∑ r = 1 10 r 2 + 1 × r ! , is equal to:

Options

  1. A11 × 11 !
  2. B10 × 11 !
  3. C11 !
  4. D101 × 10 !

Correct answer

B. 10 × 11 !

Step-by-step solution

∑ r = 1 10 r 2 + 1 r ! Let us express the r t h term, as a difference involving consecutive factorials, T r = r 2 + 1 + r - r   r !   = r 2 + r   r ! - r - 1 r ! Therefore, ⇒ T r = r r + 1 ! - r - 1 r !   Putting r = 1 , 2 , 3 , . . . , 10   and adding, we get ∑ r = 1 10 r r + 1 ! - r - 1 r !   = 10 × 11 ! ∴ ∑ r = 1 10 r 2 + 1 × r ! = 10 × 11 !

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