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Question

If [.] denotes the greatest integer function then limn[x]+[2x]+...+[nx]n2 is

A
0
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B
x
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C
x2
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D
x22
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Solution

The correct option is C x2
limn[x]+[2x]+...+[nx]n2=limn1n2nk=1[kx]

As kx1<[kx]<kx+1

nk=1(kx1)<nk=1[kx]<nk=1(kx+1)

xn(n+1)2n<nk=1[kx]<xn(n+1)2+n

x2(1+1n)1n<1n2nk=1[kx]<x2(1+1n)+1n

limn(x2(1+1n)1n)<limn1n2nk=1[kx]<limn(x2(1+1n)+1n)

x2<limn1n2nk=1[kx]<x2

limn1n2nk=1[kx]=x2

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