Question

tìm kết quả phép tính: \(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{2007}+\sqrt{2008}}\)

Answer 1

\(\frac{1}{\sqrt{n}+\sqrt{n+1}}=\sqrt{n+1}-\sqrt{n}\)

Áp dụng vô là được

Answer 2

\(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+........+\frac{1}{\sqrt{2007}+\sqrt{2008}}\)

\(=\frac{1-\sqrt{2}}{1-2}+\frac{\sqrt{2}-\sqrt{3}}{2-3}+........+\frac{\sqrt{2007}-\sqrt{2008}}{2007-2008}\)

\(=\frac{1-\sqrt{2}+\sqrt{2}-\sqrt{3}+..........+\sqrt{2007}-\sqrt{2008}}{-1}\)

\(=\frac{1-\sqrt{2008}}{-1}=\sqrt{2008-1}\)