Question

Cho \(A=\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{3}+\sqrt{4}}+...+\dfrac{1}{\sqrt{99}+\sqrt{100}}\)

CMR: \(A\notin Z\)

Answer 1

\(A< \dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{4}}+...+\dfrac{1}{\sqrt{98}+\sqrt{100}}\)

\(A< \sqrt{2}-1+\dfrac{\sqrt{4}-\sqrt{2}}{2}+\dfrac{\sqrt{6}-\sqrt{4}}{2}+...+\dfrac{\sqrt{100}-\sqrt{98}}{2}\)

\(A< \dfrac{\sqrt{100}-\sqrt{2}}{2}+\sqrt{2}-1=4+\dfrac{\sqrt{2}}{2}< 5\)

\(A>\dfrac{1}{\sqrt{1}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{5}}+...+\dfrac{1}{\sqrt{99}+\sqrt{101}}\)

\(A>\dfrac{\sqrt{3}-1+\sqrt{5}-\sqrt{3}+...+\sqrt{101}-\sqrt{99}}{2}\)

\(A>\dfrac{\sqrt{101}-1}{2}>\dfrac{\sqrt{100}-1}{2}=\dfrac{9}{2}>4\)

\(\Rightarrow4< A< 5\)