[latex]frac{1}{sqrt{1}+sqrt{2} }+frac{1}{ sqrt{2} + sqrt{3} } + frac{1}{ sqrt{3} + sqrt{4}}+....+ frac{1}{ sqrt{98} + sqrt{99} } + frac{1}{ sqrt{99}+ sqrt{100}}=[/latex] [latex]frac{sqrt{1}-sqrt{2}}{(sqrt{1}+sqrt{2})(sqrt{1}-sqrt{2}) }+frac{sqrt{2} - sqrt{3}}{( sqrt{2} + sqrt{3})(sqrt{2} - sqrt{3}) } + ....+ frac{ sqrt{98} - sqrt{99} }{( sqrt{98} + sqrt{99})( sqrt{98} - sqrt{99} ) } + frac{sqrt{99}- sqrt{100}}{ (sqrt{99}+ sqrt{100})(sqrt{99}- sqrt{100})}=[/latex] [latex]frac{sqrt{1}-sqrt{2}}{1-2}+frac{ sqrt{2} - sqrt{3} }{2-3} + frac{ sqrt{3} -sqrt{4}}{ 3-4}+....+ frac{sqrt{98} - sqrt{99} }{98-99 } + frac{sqrt{99}-sqrt{100}}{99-100}=[/latex] [latex]frac{sqrt{1}-sqrt{2}}{-1}+frac{ sqrt{2} - sqrt{3} }{-1} + frac{ sqrt{3} -sqrt{4}}{-1}+....+ frac{sqrt{98} - sqrt{99} }{-1} + frac{sqrt{99}-sqrt{100}}{-1}=[/latex] [latex]sqrt{2}-sqrt{1}+sqrt{3} - sqrt{2} + sqrt{4} -sqrt{3}+....+ sqrt{99} - sqrt{98} +sqrt{100}-sqrt{99}=[/latex] [latex]sqrt{100} - sqrt{1}=10-1=9[/latex]
