Let [ ] denote the greatest integer .

Question

Let $[\alpha ]$ denote the greatest integer $\le \alpha $. Then $[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots +[\sqrt{120}]$ is equal to \_\_\_\_\_\_\_\_\_\_

Answer: 825

Accepted Answer

$\begin{gathered} [\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots .+[120] \ E=1+1+1+2+2+2+2+2+3+3+3+3+3 \ +3+3+4+4+\ldots \ E=3	imes 1+5	imes 2+7	imes 3+\ldots .+19	imes 9+10	imes 21 \ =\sum _{r=1}^{10}(2r+1)r=2[\frac{10	imes 11	imes 21}{6}]+\frac{10	imes 11}{2} \ =770+55 \ =825 \end{gathered}$

Solutions · 1