Question

CMR \(\frac{1}{\sqrt{2}+\sqrt{1}}+\frac{1}{\sqrt{3}+\sqrt{2}}+...+\frac{1}{\sqrt{169}+\sqrt{168}}+12\)

giải giúp mk vs mai mk nộp r

Answer 1

Đặt \(A=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{168}+\sqrt{169}}\)

\(A=\frac{\sqrt{2}-\sqrt{1}}{\left(\sqrt{1}+\sqrt{2}\right)\left(\sqrt{2}-\sqrt{1}\right)}+\frac{\sqrt{3}-\sqrt{2}}{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{3}-\sqrt{2}\right)}+...+\frac{\sqrt{169}-\sqrt{168}}{\left(\sqrt{168}+\sqrt{169}\right)\left(\sqrt{169}-\sqrt{168}\right)}\)

\(A=\frac{\sqrt{2}-\sqrt{1}}{2-1}+\frac{\sqrt{3}-\sqrt{2}}{3-2}+...+\frac{\sqrt{169}-\sqrt{168}}{169-168}\)

\(A=\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+...+\sqrt{169}-\sqrt{168}\)

\(A=\sqrt{169}-\sqrt{1}\)

\(A=13-1=12\)( đpcm )