Question

The value of $1000[\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+...+\frac{1}{999\times 1000}]$ is equal to

• A. $1000$
• B. $999$
• C. $1001$
• D. $\frac{1}{999}$

Answer 1

Answer: (B)

$999$

$=1000[\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+...+\frac{1}{999\times 1000}]$

$=1000[\frac{2-1}{1\times 2}+\frac{3-2}{2\times 3}+\frac{4-3}{3\times 4}+...+\frac{1000-999}{999\times 1000}]$

$=1000[\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{999}-\frac{1}{1000}]$,

$=1000[1-\frac{1}{1000}]$ Only end terms remain, as all intermediate terms cancel out.

$=1000.\frac{999}{1000}$

$=999$