The value of expression $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + ... + [\sqrt{100}]$ (where $[.]$ denotes greatest integer function) is equal to ____.
Step-by-Step Solution
Key Concept: Difference of squares $(k+1)^2 - k^2 = 2k+1$ reduces the expression to a polynomial sum that has a closed form.
The sum $S = \sum_{k=1}^{99} k\left((k+1)^2 - k^2\right) + 100 = \sum_{k=1}^{99} k(2k+1) + 100 = \frac{100 \times 99 \times 401}{6} + 100 = 661750$. The telescoping identity $(k+1)^2 - k^2 = 2k+1$ is key, and the sum of $k(2k+1)$ yields the cubic formula.
Correct Answer: 661750