The sum $1 \times 1! + 2 \times 2! + \ldots + 50 \times 50!$ equals

If $\frac{1}{2 \times 4} + \frac{1}{4 \times 6} + \frac{1}{6 \times 8} + \dots (n \text{ terms}) = \frac{k n}{n+1}$,then $k$ is equal to

If $a_1, a_2, a_3, ..., a_n$ is an arithmetic progression,then $\frac{1}{a_1 a_2} + \frac{1}{a_2 a_3} + \frac{1}{a_3 a_4} + ... + \frac{1}{a_{n-1} a_n} = ...$

$\frac{1}{3^{2}-1}+\frac{1}{5^{2}-1}+\frac{1}{7^{2}-1}+\ldots+\frac{1}{(201)^{2}-1}$ is equal to

The sum $1(1!) + 2(2!) + 3(3!) + \dots + n(n!)$ equals