[กขฃคฅฆง]

24. $\dfrac{1}{n\sqrt{n+1}+(n+1)\sqrt{n} } = \dfrac{\sqrt{n+1}-\sqrt{n} }{n(n+1)+(n+1)\sqrt{n(n+1)}-n\sqrt{n(n+1)}-n(n+1) } = \dfrac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n(n+1)} } = \dfrac{1}{\sqrt{n} }-\dfrac{1}{\sqrt{n+1}} $

$\therefore A = \dfrac{1}{\sqrt{1} }-\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{2} }-\dfrac{1}{\sqrt{3}}+\dfrac{1}{\sqrt{3} }-\dfrac{1}{\sqrt{4}}+...+\dfrac{1}{\sqrt{528} }-\dfrac{1}{\sqrt{529}}$

$A = 1-\frac{1}{23} = \frac{22}{23} $