Q92 / 200 · WBJEE 2013 · Mathematics

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

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

Solution

$1000\left[\frac{1}{1 \times 2} \times \frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\ldots+\frac{1}{999 \times 1000}\right]$ $=1000\left\{\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)\right.$ $\left.+\ldots+\left(\frac{1}{999}-\frac{1}{1000}\right)\right]$ $=1000\left(1-\frac{1}{1000}\right)$ $=1000 \times \frac{999}{1000}$ $=999$

Hence, the correct answer is option: 'B'.

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