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JEE Main 202229 JulyEvening Shift
Exam
JEE Main
Year
2022
Date
29 July
Shift
Evening Shift
Subject
Mathematics
Chapter
Sequences and Series

JEE Main 2022 · 29 July · Evening Shift · Mathematics · Sequences and Series

∑ _ r = 1 ^ 20 ( r ^ 2 + 1) (r !) is equal to

∑r=120\mathop{∑}_{r = 1}^{20}\left({r}^{2} + 1\right)(r!)\left(r !\right) is equal to

Options

  1. A22!−21!22 ! - 21 !
  2. B22!−2(21!)22 ! - 2\left(21 !\right)
  3. C21!−2(20!)21 ! - 2\left(20 !\right)
  4. D21!−20!21 ! - 20 !

Correct answer

B. 22!−2(21!)22 ! - 2\left(21 !\right)

Step-by-step solution

Given,

∑x=120(r2+1)r!\mathop{∑}_{x = 1}^{20}\left({r}^{2} + 1\right)r !

Now on rearranging we get,

=∑x=120((r+1)2−2r)r!= \mathop{∑}_{x = 1}^{20}\left(( r + 1{)}^{2} - 2r\right)r !

=∑x=120((r+1)(r+1)!−r⋅r!)−∑r=120r⋅r!= \mathop{∑}_{x = 1}^{20}\left(\left(r + 1\right)\left(r + 1\right) ! - r \cdot r !\right) - \mathop{∑}_{r = 1}^{20}r \cdot r !

=∑x=120((r+1)(r+1)!−r.r!)−∑r=120((r+1)!−r!)= \mathop{∑}_{x = 1}^{20}\left(\left(r + 1\right)\left(r + 1\right) ! - r . r !\right) - \mathop{∑}_{r = 1}^{20}\left(\left(r + 1\right) ! - r !\right)

=(21.21−1)−(21−1)= \left(21 . 21 - 1\right) - \left(21 - 1\right)

=21.21−21= 21 . 21 - 21

=20.21!=(22−2)21!=22!−2.21!= 20 . 21 ! = \left(22 - 2\right)21 ! = 22 ! - 2 . 21 !

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