The value of 1000- 1 1- 2 - 1 2- 3 - 1 3- 4 - 1 999- 1000 - is

The correct option is C 999 1000[ 1 1× 2 + 1 2× 3 + 1 3× 4 +.....+ 1 999× 1000 #160;] =1000{( 1 1 2 #160;) +( 1 2 1 3 # ...

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Cardinality of Sets

The value of ...

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

1000

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999

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1001

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\frac{1}{999}

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Solution

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

[\frac{1}{2}] + [\frac{1}{2} + \frac{1}{1000}] + [\frac{1}{2} + \frac{2}{1000}] + [\frac{1}{2} + \frac{3}{1000}] + . . . + [\frac{1}{2} + \frac{999}{1000}] = 500

[\frac{1}{2} + \frac{r}{1000}] = {\begin{matrix} 0; & if 0 \leq r < 500 1; & if 500 \leq r \leq 999 \end{matrix}

x^{50}

(1 + x)^{1000} + 2 x (1 + x)^{999} + 3 x^{2} (1 + x)^{998} + \dots \dots \dots + 1001 x^{1000}

x^{50}

(1 + x)^{1000} + 2 x (1 + x)^{999} + 3 x^{2} (1 + x)^{998} + . . . . . . + 1001 x^{1000}

[x]

[\frac{1}{2} + \frac{1}{1000}] + [\frac{1}{2} + \frac{2}{1000}] + \dots . + [\frac{1}{2} + \frac{999}{1000}] =

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