Question

A=\(\dfrac{1}{\sqrt{1}+\sqrt{2}}\)+\(\dfrac{1}{\sqrt{2}+\sqrt{3}}\)+....+\(\dfrac{1}{\sqrt{2022}+\sqrt{2023}}\)

B=\(\dfrac{1}{\sqrt{1}}\)+\(\dfrac{1}{\sqrt{2}}\)+....+\(\dfrac{1}{\sqrt{2022}}\)

a)Rút gọn A

b)C/m B>2\(\sqrt{2023}\)-1

Answer 1

a) \(A=\dfrac{1}{1+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+...+\dfrac{1}{\sqrt{2022}+\sqrt{2023}}\)

\(=\dfrac{\sqrt{2}-1}{\left(1+\sqrt{2}\right)\left(\sqrt{2}-1\right)}+\dfrac{\sqrt{3}-\sqrt{2}}{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{3}-\sqrt{2}\right)}+...+\dfrac{\sqrt{2023}-\sqrt{2022}}{\left(\sqrt{2022}+\sqrt{2023}\right)\left(\sqrt{2023}-\sqrt{2022}\right)}\)

\(=\dfrac{\sqrt{2}-1}{2-1}+\dfrac{\sqrt{3}-\sqrt{2}}{3-2}+...+\dfrac{\sqrt{2023}-\sqrt{2022}}{2023-2022}\)

\(=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{2023}-\sqrt{2022}=\sqrt{2023}-1\)