\(\frac{1}{\sqrt{9} + \sqrt{11}} + \frac{1}{\sqrt{11} + \sqrt{13}} + \cdots + \frac{1}{\sqrt{2023} + \sqrt{2025}} \) =

Given:

\(\frac{1}{\sqrt{9} + \sqrt{11}} + \frac{1}{\sqrt{11} + \sqrt{13}} + \cdots + \frac{1}{\sqrt{2023} + \sqrt{2025}}\) = ?

Formula used:

$\frac{1}{\sqrt{a} + \sqrt{b}} = \sqrt{b} - \sqrt{a}$

Calculation:

$\frac{1}{\sqrt{9} + \sqrt{11}} + \frac{1}{\sqrt{11} + \sqrt{13}} + \cdots + \frac{1}{\sqrt{2023} + \sqrt{2025}}$

⇒ ( $\sqrt{11} - \sqrt{9}$ ) + ( $\sqrt{13} - \sqrt{11}$ ) + ... + ( $\sqrt{2025} - \sqrt{2023}$ )

⇒ - $\sqrt{9}$ + $\sqrt{2025}$

$\sqrt{9} = 3$ , $\sqrt{2025} = 45$

⇒ 45 - 3 = 42

∴ The correct answer is option (4).