First slide

Series of natural numbers

Question

If $[x]$ denotes the greatest integer $\leq x,$ then value of $S = [\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + \dots + [\sqrt{2024}]$ is

Moderate

A

$59000$
B

$58750$
C

$59730$
D

$65138$

Solution

For any positive integer $k$ and $x$ , we have
$[\sqrt{x}] = k \Leftrightarrow k^{2} \leq x < (k + 1)^{2} \Leftrightarrow x \in [k^{2}, k^{2} + 2 k)$
$∴ [\sqrt{x}] = 1$ for $x = 1, 2, 3 .$
$[\sqrt{x}] = 2$ for $x = 4, 5, 6, 7, 8,$
$[\sqrt{x}] = 3$ for $x = 9, 10, 11, 12, 13, 14, 15,$
$. . . . . .$
$[\sqrt{x}] = 44$ for $x = 1936, . . . . . . ., 2024$
Thus, $S = (1) (3) + (2) (5) + 3 (7) + \dots + 44 (89)$
$\begin{array}{l} = \sum_{k = 1}^{44} (k) (2 k + 1) = 2 \sum_{k = 1}^{44} (k)^{2} + \sum_{k = 1}^{44} (k) \\ = \frac{2}{6} (44) (45) (89) + \frac{1}{2} (44) (45) \\ = 59730 \end{array}$

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