QJEEM23Ma5953: JEE Main 2023 Mathematics - Class 11 JEEM23Ma5953 Solution: Let denote the greatest integer . Then is equal to __________

Question

Verified Solution • 2025–26 Session moderate

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

Accepted Answer

\begin{aligned}&amp; {[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots .+[120]} \\\\&amp; E=1+1+1+2+2+2+2+2+3+3+3+3+3 \\\\&amp; +3+3+4+4+\ldots \\\\&amp; E=3 \times 1+5 \times 2+7 \times 3+\ldots .+19 \times 9+10 \times 21 \\\\&amp; =\sum_{r=1}^{10}(2 r+1) r=2\left[\frac{10 \times 11 \times 21}{6}\right]+\frac{10 \times 11}{2} \\\\&amp; =770+55 \\\\&amp; =825 \\\\&amp;\end{aligned}

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