The value of (1/2!)+(2/3!)+....+(999/1000!) is equal to

Question

The value of (1/2!)+(2/3!)+....+(999/1000!) is equal to (A)  (1000!-1/1000!)  (B)  (1000!+1/1000!)  (C)  (999!-1/999!)  (D)  (999!+1/999!)

Tardigrade Admin · Accepted Answer

(1/2!)+(2/3!)+....+(999/1000!) =(2-1/2!)+(3-1/3!)+.....+(1000-1/1000!) =(1/1!)+(1/2!)+(1/2!)-(1/3!)+....+(1/999!)-(1/1000!) =1-(1/1000!)=(1000!-1/1000!)

Q. The value of $\frac{1}{2 !} + \frac{2}{3 !} + .... + \frac{999}{1000 !}$ is equal to

$\frac{1000 ! - 1}{1000 !}$

$\frac{1000 ! + 1}{1000 !}$

$\frac{999 ! - 1}{999 !}$

$\frac{999 ! + 1}{999 !}$

$\frac{1000 ! - 999 !}{1000 !}$

$\frac{1}{2 !} + \frac{2}{3 !} + .... + \frac{999}{1000 !}$
$= \frac{2 - 1}{2 !} + \frac{3 - 1}{3 !} + ..... + \frac{1000 - 1}{1000 !}$
$= \frac{1}{1 !} + \frac{1}{2 !} + \frac{1}{2 !} - \frac{1}{3 !} + .... + \frac{1}{999 !} - \frac{1}{1000 !}$
$= 1 - \frac{1}{1000 !} = \frac{1000 ! - 1}{1000 !}$

Q. The value of 2!1​+3!2​+....+1000!999​ is equal to

1000!1000!−1​

1000!1000!+1​

999!999!−1​

999!999!+1​

1000!1000!−999!​

2!1​+3!2​+....+1000!999​ =2!2−1​+3!3−1​+.....+1000!1000−1​ =1!1​+2!1​+2!1​−3!1​+....+999!1​−1000!1​ =1−1000!1​=1000!1000!−1​