Question

So sánh các số sau:

A=\(2\sqrt{1}+2\sqrt{3}+2\sqrt{5}+...+2\sqrt{19}\)

B=\(2\sqrt{2}+2\sqrt{4}+2\sqrt{6}+...+2\sqrt{18}+\sqrt{20}\)

Answer 1

Ta có : \(\sqrt{3}-\sqrt{2}=\dfrac{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}{\sqrt{3}+\sqrt{2}}=\dfrac{1}{\sqrt{3}+\sqrt{2}}>\dfrac{1}{\sqrt{4}+\sqrt{3}}=\sqrt{4}-\sqrt{3}\Rightarrow\sqrt{3}-\sqrt{2}>\sqrt{4}-\sqrt{3}\Rightarrow2\sqrt{3}>\sqrt{4}+\sqrt{2}\)

Làm tương tự : \(2\sqrt{5}>\sqrt{4}+\sqrt{6};2\sqrt{7}>\sqrt{6}+\sqrt{8},...,2\sqrt{19}>\sqrt{18}+\sqrt{20}\)

Cộng từng BĐT trên , ta được :

\(2\sqrt{3}+2\sqrt{5}+...+2\sqrt{19}>\sqrt{4}+\sqrt{2}+\sqrt{4}+\sqrt{6}+...+\sqrt{18}+\sqrt{20}=2\sqrt{4}+2\sqrt{6}+...+2\sqrt{18}+\sqrt{20}+\sqrt{2}\)

\(\Leftrightarrow A-2\sqrt{1}>B-\sqrt{2}\)

\(\Leftrightarrow A-B>2-\sqrt{2}>0\Rightarrow A>B\)