Bonjour,
1)
[tex]\dfrac{1}{\sqrt{n+1}+\sqrt{n}} \\\\\\=\dfrac{\sqrt{n+1}-\sqrt{n}}{(\sqrt{n+1}-\sqrt{n})\sqrt{n+1}+\sqrt{n})}\\\\\\=\dfrac{\sqrt{n+1}-\sqrt{n}}{1}\\\\=\sqrt{n+1}-\sqrt{n}[/tex]
2)
[tex]\displaystyle S=\sum_{i=1}^{99}(\dfrac{1}{\sqrt{i}+\sqrt{i+1}} )\\\\=\sum_{i=1}^{99}(\dfrac{1}{\sqrt{i+1}+\sqrt{i}} )\\\\=\sum_{i=1}^{99}(\sqrt{i+1}-\sqrt{i} )\\\\=(\sqrt{2}-\sqrt{1}) +(\sqrt{3}-\sqrt{2} )+(\sqrt{4}-\sqrt{3}) +....+(\sqrt{99}-\sqrt{98} )+(\sqrt{100}-\sqrt{99} )\\\\=-\sqrt{1}+\sqrt{100} \\\\=9\\[/tex]
