Question

so sánh \(\sqrt{2005}-\sqrt{2004}\) và \(\sqrt{2004}-\sqrt{2003}\)

Answer 1

Ta có: +) 2005 - 2004 = 1 \(\Leftrightarrow\left(\sqrt{2005}-\sqrt{2004}\right)\left(\sqrt{2005}+\sqrt{2004}\right)=1\)

\(\Leftrightarrow\sqrt{2005}-\sqrt{2004}=\dfrac{1}{\sqrt{2005}+\sqrt{2004}}\) (1)

+) \(2004-2003=1\)

\(\Leftrightarrow\left(\sqrt{2004}-\sqrt{2003}\right)\left(\sqrt{2004}+\sqrt{2003}\right)=1\)

\(\Leftrightarrow\sqrt{2004}-\sqrt{2003}=\dfrac{1}{\sqrt{2004}+\sqrt{2003}}\) (2)

Vì \(\sqrt{2005}+\sqrt{2004}>\sqrt{2004}+\sqrt{2003}\)

\(\Rightarrow\dfrac{1}{\sqrt{2005}+\sqrt{2004}}< \dfrac{1}{\sqrt{2004}+\sqrt{2003}}\) (3)

Từ (1), (2) và (3) \(\Rightarrow\sqrt{2005}-\sqrt{2004}< \sqrt{2004}-\sqrt{2003}\)