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Mathematics

Functions

Easy

Question

If x and {x} represents integral and fractional parts of x,then $left square bracket x right square bracket plus sum from r equals 1 to 2000 of fraction numerator left curly bracket x plus r right curly bracket over denominator 2000 end fraction equals$

$\frac{2001}{2} x$
$x + 2001$
$x$
$[x] + \frac{2001}{2}$

The correct answer is: $x$

$r text is a positive integer end text not stretchy rightwards double arrow left curly bracket x plus r right curly bracket equals left curly bracket x right curly bracket therefore left square bracket x right square bracket plus sum from r equals 1 to 2000 of fraction numerator left curly bracket x plus r right curly bracket over denominator 2000 end fraction equals left square bracket x right square bracket plus sum from r equals 1 to 2000 of fraction numerator left curly bracket x right curly bracket over denominator 2000 end fraction equals left square bracket x right square bracket plus fraction numerator left curly bracket x right curly bracket over denominator 2000 end fraction cross times 2000 equals left square bracket x right square bracket plus left curly bracket x right curly bracket equals x$

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