[tex]\dfrac{1}{\sqrt 5+\sqrt 3}+\dfrac{1}{\sqrt 7+\sqrt 5}+...+\dfrac{1}{\sqrt{99}+\sqrt{97}} = \\ \\ = \displaystyle \sum\limits_{k=2}^{49}\dfrac{1}{\sqrt{2k+1}+\sqrt{2k-1}} =\\ \\ = \sum\limits_{k=2}^{49}\dfrac{\sqrt{2k+1}-\sqrt{2k-1}}{(\sqrt{2k+1}-\sqrt{2k-1})(\sqrt{2k+1}+\sqrt{2k-1})} = \\ \\ = \sum\limits_{k=2}^{49}\dfrac{\sqrt{2k+1}-\sqrt{2k-1}}{2k+1-(2k-1)} =\sum\limits_{k=2}^{49}\dfrac{\sqrt{2k+1}-\sqrt{2k-1}}{2} =[/tex]
[tex]=\dfrac{1}{2}\Big(\sqrt 5+\sqrt 7+\sqrt 9 +...+\sqrt{99} - \sqrt{3} - \sqrt 5 - \sqrt 7 - ... - \sqrt{97}\Big) = \\ \\ = \dfrac{1}{2}\Big(\sqrt{99}-\sqrt{3}\Big) = \dfrac{3\sqrt{11} - \sqrt 3}{2}[/tex]
