Question

$\frac{10001\times 100!}{2\times 1!+5\times 2!+10\times 3!+\dots +10001\times 100!}=$

• A. $\frac{1001}{1100}$
• B. $\frac{10001}{10100}$
• C. $\frac{101}{110}$
• D. $\frac{100001}{101000}$

Answer 1

Answer: (B)

Here, general term of denominator is
$\sum _{n=1}^{100}\left(n^2+1\right)n!=\sum _{n=1}^{100}\left[(n+1{)}^{2}n!-2n,n!\right]$
$=\sum _{n=1}^{100}[(n+1)(n+1)!-n\cdot n!]-\sum _{n=1}^{100}n\cdot n!$
$=\sum _{n=1}^{100}[(n+1)(n+1)!-n\cdot n!]-\sum _{n=1}^{100}(n+1)!-n!$
$\left[\begin{array}{ccc}2\cdot 2!& -& 1\cdot 1!\\ +3\cdot 3!& -& 2\cdot 2!\\ ⋮& & \\ +101\cdot 101!& -& 100\cdot 100!\end{array}\right]-\left[\begin{array}{ccc}2!& -& 1!\\ +3!& -& 2!\\ ⋮& & \\ +101!& -& 100!\end{array}\right]$
$=[101\cdot 101!-1]-[101!-1]$
Denominator $=100\cdot 101!$
Consider, $\frac{10001\times 100!}{100\times 101\times 100!}$
$=\frac{10001}{10100}$