Let [α] denote the greatest integer ≤ α . Then [ 1 ] + [ 2 ] + [ 3...

Given,

$\left[\sqrt{1}\right] + \left[\sqrt{2}\right] + \left[\sqrt{3}\right] + . . . . . . . . . . . . . + \left[\sqrt{120}\right]$

Now we now that,

\left[\sqrt{x}\right] = 1, \text{when}x \in [ 1, 4 )$, \left[\sqrt{x}\right] = 2, \text{when}x \in [ 4, 9 )$ and similarly \left[\sqrt{x}\right] = 10, \text{when}x \in [ 100, 120 )$</p><p>Now using the above formula we get,</p><p>$E = 1 + 1 + 1 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 4 + 4 + \ldots$</p><p>$⇒ E = 3 \times 1 + 5 \times 2 + 7 \times 3 + \ldots . . + 19 \times 9 + 10 \times 21$</p><p>$⇒ E = \mathop{∑}_{r = 1}^{10}\left(2r + 1\right)r$</p><p>$⇒ E = 2\mathop{∑}_{r = 1}^{10}\left({r}^{2} + r\right)$</p><p>$⇒ E = 2\left[\dfrac{10 \times 11 \times 21}{6}\right] + \dfrac{10 \times 11}{2}$</p><p>$⇒ E = 770 + 55$</p><p>$⇒ E = 825