A) \[\frac{1000!-1}{1000!}\]
B) \[\frac{1000!+1}{1000!}\]
C) \[\frac{999!-1}{999!}\]
D) \[\frac{999!+1}{999!}\]
E) \[\frac{1000!-999!}{1000!}\]
Correct Answer: A
Solution :
\[\frac{1}{2!}+\frac{2}{3!}+....+\frac{999}{1000!}\] \[=\frac{2-1}{2!}+\frac{3-1}{3!}+.....+\frac{1000-1}{1000!}\] \[=\frac{1}{1!}+\frac{1}{2!}+\frac{1}{2!}-\frac{1}{3!}+....+\frac{1}{999!}-\frac{1}{1000!}\] \[=1-\frac{1}{1000!}=\frac{1000!-1}{1000!}\]
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