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(D) Let the given sum be $S = \sum_{k=0}^{99} \left[\frac{1}{2} + \frac{k}{100}\right]$.We observe the value of each term $\left[\frac{1}{2} + \frac{k

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$50$

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(D) Let the given sum be $S = \sum_{k=0}^{99} \left[\frac{1}{2} + \frac{k}{100}ight]$.We observe the value of each term $\left[\frac{1}{2} + \frac{k}{100}ight]$:For $0 \le k \le 49$,we have $\frac{1}{2} \le \frac{1}{2} + \frac{k}{100} \le \frac{1}{2} + \frac{49}{100} = 0.99$. Since $0 \le 0.5 + \frac{k}{100} For $50 \le k \le 99$,we have $\frac{1}{2} + \frac{50}{100} \le \frac{1}{2} + \frac{k}{100} \le \frac{1}{2} + \frac{99}{100}$,which simplifies to $1 \le \frac{1}{2} + \frac{k}{100} \le 1.49$. Since $1 \le 0.5 + \frac{k}{100} Thus,$S = (50 	imes 0) + (50 	imes 1) = 50$.

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