Question:
If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of $\sum_{r=1}^{100} \left[ \frac{r}{5} \right]$ is:
If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of $\sum_{r=1}^{100} \left[ \frac{r}{5} \right]$ is:
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Recognize the periodic nature of division steps: $[r/d]$ repeats $d$ times for each integer value except at the boundaries. Here, there are $5$ copies of each integer from $1$ to $19$, with a single $20$ at the end.
Updated On: May 31, 2026
- $980$
- $950$
- $960$
- $970$
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The Correct Option is D
Solution and Explanation
Step 1: Concept
The greatest integer function $[x]$ increases by 1 for every unit interval. For $\left[\frac{r}{5}\right]$, the value is a constant integer for every group of 5 consecutive terms of $r$.
Step 2: Meaning
We can group the terms from $r = 1$ to $100$ into blocks of size 5 to simplify the summation.
Step 3: Analysis
Evaluating the terms systematically:
• For $r = 1, 2, 3, 4$, $\left[\frac{r}{5}\right] = 0$ (4 terms)
• For $r = 5, 6, 7, 8, 9$, $\left[\frac{r}{5}\right] = 1$ (5 terms)
• For $r = 10, 11, 12, 13, 14$, $\left[\frac{r}{5}\right] = 2$ (5 terms)
• $\dots$
• For $r = 95, 96, 97, 98, 99$, $\left[\frac{r}{5}\right] = 19$ (5 terms)
• For $r = 100$, $\left[\frac{100}{5}\right] = 20$ (1 term) Summing these up: \[ S = (0 \times 4) + (1 \times 5) + (2 \times 5) + \dots + (19 \times 5) + 20 \] \[ S = 5 \times (1 + 2 + \dots + 19) + 20 \] Using the sum of first $n$ natural numbers formula: \[ 1 + 2 + \dots + 19 = \frac{19 \times 20}{2} = 190 \] \[ S = 5 \times 190 + 20 = 950 + 20 = 970 \]
Step 4: Conclusion
The final sum of the given terms is $970$. Final Answer: (D)
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