First slide

Permutations

Question

For x ∈ R, let [x] denotes the greatest integer ≤ x, then the value of $[- \frac{1}{3}] + [- \frac{1}{3} - \frac{1}{100}] + [- \frac{1}{3} - \frac{2}{100}] + \dots + [- \frac{1}{3} - \frac{99}{100}]$ is

Moderate

A

–100
B

–123
C

–135
D

–153

Solution

For $0 \leq r \leq 66, 0 \leq \frac{r}{100} < \frac{2}{3}$
$\begin{array}{l} \Rightarrow - \frac{2}{3} < - \frac{r}{100} \leq 0 \\ \Rightarrow - \frac{1}{3} - \frac{2}{3} < - \frac{1}{3} - \frac{r}{100} \leq - \frac{1}{3} \\ ∴ [- \frac{1}{3} - \frac{r}{100}] = - 1 for 0 \leq r \leq 66 \end{array}$
Also, for $67 \leq r \leq 100, \frac{67}{100} \leq \frac{r}{100} \leq 1$
$\begin{array}{l} \Rightarrow - 1 \leq - \frac{r}{100} \leq - \frac{67}{100} \\ \Rightarrow - \frac{1}{3} - 1 \leq - \frac{1}{3} - \frac{r}{100} \leq - \frac{1}{3} - \frac{67}{100} \\ ∴ [- \frac{1}{3} - \frac{r}{100}] = - 2 for 67 \leq r \leq 100 \end{array}$
Hence, $\sum_{r = 0}^{100} [- \frac{1}{3} - \frac{r}{100}] = 67 (- 1) + 2 (- 34) = - 135$ .

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