_ r = 0 ^ 100 ( r^2 + 4r + 4) | ! , r + 1 , . is equal to :- - Vidyadip

MCQ

$\sum\limits_{r = 0}^{100} {({r^2} + 4r + 4)\left| \!{\underline {\,
{r + 1} \,}} \right. } $ is equal to :-

✓
$\left| \!{\underline {\,
{103} \,}} \right. - 2$
B
$\left| \!{\underline {\,
{102} \,}} \right. - 1$
C
${\left( {\left| \!{\underline {\,
{102} \,}} \right. } \right)^2}$
D
None

✓

Answer

Correct option: A.

$\left| \!{\underline {\,
{103} \,}} \right. - 2$

a
$\sum\limits_{r = 0}^{100} {{{\left( {r + 2} \right)}^2}r + 1!} = \sum\limits_{r = 0}^{100} {\left( {r + 2} \right)} \,r + 2!$

$\sum\limits_{r = 0}^{100} {\left( {r + 3 - 1} \right)} \,r + 2! = \sum\limits_{r = 0}^{100} {r + 3!} - \sum\limits_{r = 0}^{100} {r + 2!} $

$ \Rightarrow \left( {3! + 4! + 5! + ... + 103!} \right) - \left( {2! + 3! + 4! + ... + 102!} \right)$

$ \Rightarrow 103! - 2$

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