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The value of $1000\left[\frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\ldots+\frac{1}{999 \times 1000}\right]$ is
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The correct answer is:
999
$1000\left[\frac{1}{1 \times 2} \times \frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\ldots+\frac{1}{999 \times 1000}\right]$
$=1000\left\{\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)\right.$
$\left.+\ldots+\left(\frac{1}{999}-\frac{1}{1000}\right)\right]$
$=1000\left(1-\frac{1}{1000}\right)$
$=1000 \times \frac{999}{1000}$
$=999$
$=1000\left\{\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)\right.$
$\left.+\ldots+\left(\frac{1}{999}-\frac{1}{1000}\right)\right]$
$=1000\left(1-\frac{1}{1000}\right)$
$=1000 \times \frac{999}{1000}$
$=999$
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