Answer:
Hello,
Step-by-step explanation:
[tex]\dfrac{1}{\sqrt{1} +\sqrt{2} } =\dfrac{\sqrt{2} -\sqrt{1}}{(\sqrt{2} +\sqrt{1})*(\sqrt{2} -\sqrt{1}) } =\dfrac{\sqrt{2} -1}{2-1} =\sqrt{2} -1\\\\\dfrac{1}{\sqrt{2} +\sqrt{3} } =\dfrac{\sqrt{3} -\sqrt{2}}{(\sqrt{3} +\sqrt{2})*(\sqrt{3} -\sqrt{2}) } =\dfrac{\sqrt{3} -\sqrt{2}}{3-2} =\sqrt{3} -\sqrt{2}\\\\\\\dfrac{1}{\sqrt{n-1} +\sqrt{n} } =\dfrac{\sqrt{n} -\sqrt{n-1}}{(\sqrt{n} +\sqrt{n-1})*(\sqrt{n} -\sqrt{n-1}) }=\sqrt{n} -\sqrt{n-1}\\\\[/tex]
[tex]\\\dfrac{1}{\sqrt{1} +\sqrt{2} } +\dfrac{1}{\sqrt{2} +\sqrt{3} } +...+\dfrac{1}{\sqrt{n-1} +\sqrt{n} } =\boxed{\sqrt{n} -1}\\[/tex]
