Question

\(Rútgọn\):

\(a,\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{48}+\sqrt{49}}\)

\(b,\frac{1}{\sqrt{3}-\sqrt{5}}+\frac{1}{\sqrt{3}+\sqrt{5}}\)

Answer 1

a) \(\frac{1}{1+\sqrt{2}}\)+\(\frac{1}{\sqrt{2}+\sqrt{3}}\)+\(\frac{1}{\sqrt{3}+\sqrt{4}}\)+...+\(\frac{1}{\sqrt{48}+\sqrt{49}}\)

=\(\frac{\sqrt{2}-1}{\left(\sqrt{2}+1\right).\left(\sqrt{2}-1\right)}\)+\(\frac{\sqrt{3}-\sqrt{2}}{\left(\sqrt{3}+\sqrt{2}\right).\left(\sqrt{3}-\sqrt{2}\right)}+...+\)\(\frac{\sqrt{49}-\sqrt{48}}{\left(\sqrt{49}+\sqrt{48}\right).\left(\sqrt{49}-\sqrt{48}\right)}\)

=\(\frac{\sqrt{2}-1}{2-1}\)+\(\frac{\sqrt{3}-\sqrt{2}}{3-2}\)+...+\(\frac{\sqrt{49}-\sqrt{48}}{49-48}\)

=\(\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\)\(\sqrt{49}-\sqrt{48}\)

=\(\sqrt{49}-1\)=\(7-1\)= \(6\)