Question

Tính giá trị biểu thức:

\(M=\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+...+\frac{1}{25\sqrt{24}+24\sqrt{25}}\)

Answer 1

Có \(\frac{1}{n\sqrt{n-1}+\left(n-1\right)\sqrt{n}}\left(n\ge2\right)=\frac{1}{\sqrt{n}.\sqrt{n-1}\left(\sqrt{n}+\sqrt{n-1}\right)}=\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n}.\sqrt{n-1}}=\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt{n}}\)

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

Áp dụng (1) vào bt M có:

M=\(1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{24}}-\frac{1}{\sqrt{25}}\)=\(1-\frac{1}{5}\)=\(\frac{4}{5}\)

Vậy M=\(\frac{4}{5}\)