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Greatest Binomial Coefficients

f x = x + x +...

Question

f(x) = {x} + {x + 1} + {x + 2} + .......+ {x + 999} then $[f (\sqrt{2})]$ (where {.} denotes fractional part of x and [.] denotes greatest integer of x) is equal to

999 \times 500

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414

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4140

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51010

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Solution

The correct option is B $414$
${x + r} = {x}$ if r is integer
$x + r - [x] - r = x - [x]$
$\Rightarrow f (x) = 1000 {x}$
$[f (\sqrt{2})] = [100 \times 0.414] = 414$

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