Răspuns:
9
Explicație pas cu pas:
Amplificam cu conjugata expresiei de la numitor iar apoi folosim formula
(a+b)(a-b)=a²-b²
[tex]\frac{1}{1+\sqrt{2} } +\frac{1}{\sqrt{2}+\sqrt{3} } +...+\frac{1}{\sqrt{99} +\sqrt{100} } =\\ \\ \frac{\sqrt{2}-1}{(\sqrt{2}-1)(\sqrt{2}+1) } +\frac{\sqrt{3}-\sqrt{2} }{(\sqrt{3}-\sqrt{2)(\sqrt{3}+\sqrt{2} } } +...+\frac{\sqrt{100}-\sqrt{99} }{(\sqrt{100}-\sqrt{99})(\sqrt{100}+\sqrt{99)} } =\\ \\ \frac{\sqrt{2}-1 }{1} +\frac{\sqrt{3} -\sqrt{2} }{1} +...+\frac{\sqrt{100}-\sqrt{99} }{1} =\\ \\ \sqrt{2} -1+\sqrt{3} -\sqrt{2} +...+\sqrt{100} -\sqrt{99} = -1+10=9[/tex]
