Question

So sánh

\(A=\dfrac{1}{1+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+...+\dfrac{1}{\sqrt{79}+\sqrt{80}}\) và \(B=\dfrac{1}{\sqrt{2}+\sqrt{3}}+\dfrac{1}{\sqrt{4}+\sqrt{5}}+...+\dfrac{1}{\sqrt{80}+\sqrt{81}}\)

Answer 1

Lời giải:

Ta thấy:

\(A-B=\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+..+\frac{1}{\sqrt{79}+\sqrt{80}}-\left(\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{4}+\sqrt{5}}+...+\frac{1}{\sqrt{80}+\sqrt{81}}\right)\)

\(=\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{5}+\sqrt{6}}+...+\frac{1}{\sqrt{79}+\sqrt{80}}-\frac{1}{\sqrt{80}+\sqrt{81}}\)

\(> \frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{77}+\sqrt{78}}>0\)

\(\Rightarrow A>B\)