Question

Tính GTBT 

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

Answer 1

gyhhunjyvjvu87r6t797787o809988 809

Answer 2

Xét \(\frac{1}{\left(k+1\right)\sqrt{k}+k\sqrt{k+1}}=\frac{1}{\sqrt{k\left(k+1\right)\left(\sqrt{k}+\sqrt{k+1}\right)}}\)

                                                   \(=\frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k\left(k+1\right)}\left(k+1-k\right)}\)

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

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

             \(=1-\frac{1}{10}=\frac{9}{10}\)