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Let \([x]\) denote the greatest integer not exceeding \(x\) and {\(x\)} \(= x - [x]\). If \(n\) is a natural number, then the sum of all values of \(x\) satisfying the equation \(2[x] = x +…

Question and options

Let [x][x] denote the greatest integer not exceeding xx and {xx} =x−[x]= x - [x]. If nn is a natural number, then the sum of all values of xx satisfying the equation 2[x]=x+n2[x] = x + n{xx} is
  1. n(n+2)2\dfrac{n(n+2)}{2}
  2. 32\dfrac{3}{2}
  3. n(n+1)2\dfrac{n(n+1)}{2}
  4. nn

Answer and explanation

Correct answer: A - \(\dfrac{n(n+2)}{2}\)
Write x=I+f. 2I = I+f+nf gives I=(n+1)f, so f=k/(n+1), I=k for k=0,1,...,n (f<1). x = k(n+2)/(n+1); summing k=0..n gives (n+2)/(n+1) * n(n+1)/2 = n(n+2)/2.

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