Question

\(\frac{1}{1+\sqrt{2}}+\frac{2}{\sqrt{2}+\sqrt{3}}+......+\frac{1}{\sqrt{99}+\sqrt{100}}\)

Answer 1

Sửa đề: \(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{99}+\sqrt{100}}\)

Ta có: \(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{99}+\sqrt{100}}\)

\(=\frac{\sqrt{2}-1}{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}+\frac{\sqrt{3}-\sqrt{2}}{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}+...+\frac{10-\sqrt{99}}{\left(10+\sqrt{99}\right)\left(10-\sqrt{99}\right)}\)

\(=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+10-\sqrt{99}\)

\(=-1+10=9\)