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:
Show Hint
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.
Step 1: Meaning of the greatest integer. The symbol $[x]$ means the largest whole number that is not bigger than $x$. So $[r/5]$ stays the same for several values of $r$ in a row.
Step 2: See the blocks. For $r=1$ to $4$ the value is $0$. For $r=5$ to $9$ it is $1$. For $r=10$ to $14$ it is $2$, and so on. Each full block has 5 numbers.
Step 3: Reach the top. The last full block is $r=95$ to $99$ giving value $19$. Then $r=100$ alone gives $[100/5]=20$.
Step 4: Write the sum. The first block of 0 adds nothing. Then \[ S = 5(1 + 2 + \cdots + 19) + 20 \] because each value from 1 to 19 appears 5 times.
Step 5: Add 1 to 19. Using $1+2+\cdots+n = \frac{n(n+1)}{2}$: \[ 1+2+\cdots+19 = \frac{19 \times 20}{2} = 190 \]
Step 6: Final total. So \[ S = 5 \times 190 + 20 = 950 + 20 = 970 \] \[ \boxed{ 970 } \]