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Ta có : \(\frac{2008}{\sqrt{2009}}+\frac{2009}{\sqrt{2008}}=\frac{2009-1}{\sqrt{2009}}+\frac{2008+1}{\sqrt{2008}}=\sqrt{2009}+\sqrt{2008}+\left(\frac{1}{\sqrt{2008}}-\frac{1}{\sqrt{2009}}\right)\)
Vì \(\frac{1}{\sqrt{2008}}>\frac{1}{\sqrt{2009}}\) nên \(\frac{1}{\sqrt{2008}}-\frac{1}{\sqrt{2009}}>0\)
\(\Rightarrow\sqrt{2009}+\sqrt{2008}+\left(\frac{1}{\sqrt{2008}}-\frac{1}{\sqrt{2009}}\right)>\sqrt{2009}+\sqrt{2008}\)
Hay \(\frac{2008}{\sqrt{2009}}+\frac{2009}{\sqrt{2008}}>\sqrt{2008}+\sqrt{2009}\)
Ta có :
\(\frac{2008}{\sqrt{2009}}+\frac{2009}{\sqrt{2008}}=\frac{2009}{\sqrt{2009}}-\frac{1}{\sqrt{2009}}+\frac{2008}{\sqrt{2008}}+\frac{1}{\sqrt{2008}}\)
\(=\sqrt{2008}+\sqrt{2009}+\frac{1}{\sqrt{2008}}-\frac{1}{\sqrt{2009}}\)
Mà \(\sqrt{2008}< \sqrt{2009}\Rightarrow\frac{1}{\sqrt{2008}}>\frac{1}{\sqrt{2009}}\Leftrightarrow\frac{1}{\sqrt{2008}}-\frac{1}{\sqrt{2009}}\)
\(\Leftrightarrow\sqrt{2008}+\sqrt{2009}+\frac{1}{\sqrt{2008}}-\frac{1}{\sqrt{2009}}>\sqrt{2008}+\sqrt{2009}\)
⇒ đpcm
so sánh \(\frac{2008}{\sqrt[]{2009}}+\frac{2009}{\sqrt[]{2008}}\) và \(\sqrt[]{2008}+\sqrt[]{2009}\)
Ta có : \(\frac{2008}{\sqrt{2009}}+\frac{2009}{\sqrt{2008}}\) = \(\frac{2009-1}{\sqrt{2009}}+\frac{2008+1}{\sqrt{2008}}\)
= \(\frac{2009}{\sqrt{2009}}-\frac{1}{\sqrt{2009}}+\frac{2008}{\sqrt{2008}}+\frac{1}{\sqrt{2008}}\)
= \(\frac{\left(\sqrt{2009}\right)^2}{\sqrt{2009}}-\frac{1}{\sqrt{2009}}+\frac{\left(\sqrt{2008}\right)^2}{\sqrt{2008}}+\frac{1}{\sqrt{2008}}\)
= \(\sqrt{2009}-\frac{1}{\sqrt{2009}}+\sqrt{2008}+\frac{1}{\sqrt{2008}}\)
Mà \(\frac{1}{\sqrt{2008}}>\frac{1}{\sqrt{2009}}\)
=> \(\frac{1}{\sqrt{2008}}-\frac{1}{\sqrt{2009}}>0\)
=> \(\frac{1}{\sqrt{2008}}-\frac{1}{\sqrt{2009}}+\sqrt{2008}+\sqrt{2009}>\sqrt{2008}+\sqrt{2009}\)
Vậy \(\frac{1}{\sqrt{2008}}-\frac{1}{\sqrt{2009}}+\sqrt{2008}+\sqrt{2009}>\sqrt{2008}+\sqrt{2009}\) .
\(C=\frac{\frac{1}{2008}-\frac{1}{2009}-\frac{1}{2010}}{\frac{5}{2008}-\frac{5}{2009}-\frac{5}{2010}}+\frac{\frac{2}{2007}-\frac{2}{2008}-\frac{2}{2009}}{\frac{3}{2007}-\frac{3}{2008}-\frac{3}{2009}}\)
\(=\frac{\frac{1}{2008}-\frac{1}{2009}-\frac{1}{2010}}{5.\left(\frac{1}{2008}-\frac{1}{2009}-\frac{1}{2010}\right)}+\frac{2.\left(\frac{1}{2007}-\frac{1}{2008}-\frac{1}{2009}\right)}{3.\left(\frac{1}{2007}-\frac{1}{2008}-\frac{1}{2009}\right)}\)
\(=\frac{1}{5}+\frac{2}{3}\)
\(=\frac{13}{15}\)
Ta có : \(\frac{a_1}{a_2}=\frac{a_2}{a_3}=\frac{a_3}{a_4}=...=\frac{a_{2008}}{a_{2009}}=\frac{a_1+a_2+a_3+...+a_{2008}}{a_2+a_3+a_4+...+a_{2009}}\)
Đặt \(\frac{a_1+a_2+a_3+...+a_{2008}}{a_2+a_3+a_4+...+a_{2009}}=b\)thì \(\frac{a_1}{a_2}=b\left(1\right);\frac{a_2}{a_3}=b\left(2\right);\frac{a_3}{a_4}=b\left(3\right);...;\frac{a_{2008}}{a_{2009}}=b\left(2008\right)\)
Nhân (1),(2),(3),...,(2008) vế theo vế,ta có :
\(\frac{a_1}{a_2}.\frac{a_2}{a_3}.\frac{a_3}{a_4}.....\frac{a_{2008}}{a_{2009}}=b^{2008}\)hay \(\frac{a_1}{a_{2009}}=\left(\frac{a_1+a_2+a_3+...+a_{2008}}{a_2+a_3+a_4+...+a_{2009}}\right)^{2008}\)(đpcm)
\(\frac{a_1}{a_2}=\frac{a_2}{a_3}=....=\frac{a_{2008}}{a_{2009}}=\frac{a_1+a_2+....+a_{2008}}{a_2+a_3+....+a_{2009}}\)
=> \(\left(\frac{a_1}{a_2}\right)^{2008}=\left(\frac{a_2}{a_3}\right)^{2008}=....=\left(\frac{a_{2008}}{a_{2009}}\right)^{2008}=\left(\frac{a_1+a_2+....+a_{2008}}{a_2+a_3+....+a_{2009}}\right)^{2008}\)
\(=\frac{a_1.a_2....a_{2008}}{a_2.a_3....a_{2009}}=\frac{a_1}{a_{2009}}\)
=> \(\left(\frac{a_1+a_2+....+a_{2008}}{a_2+a_3+....+a_{2009}}\right)^{2008}=\frac{a_1}{a_{2009}}\)
=> Đpcm
ta có: \(A=\frac{2006}{2007}+\frac{2007}{2008}+\frac{2008}{2009}+\frac{2009}{2006}\)
A = \(1-\frac{1}{2007}+1-\frac{1}{2008}+1-\frac{1}{2009}+1+\frac{3}{2006}\)
A= \(4\)\(+\frac{3}{2006}-\left(\frac{1}{2007}+\frac{1}{2008}+\frac{1}{2009}\right)\)
Do 1/2007 < 1/2006 ; 1/2008<1/2006 ; 1/2009<1/2006=> 1/2007 + 1/2008 + 1/2009 < 1/2006 + 1/2006 + 1/2006
Mà 1/2006 + 1/2006 + 1/2006 = 3/2006
=> 3/2006 -( 1/2007 + 1/2008 + 1/2009) > 0
=> \(4+\frac{3}{2006}-\left(\frac{1}{2007}+\frac{1}{2008}+\frac{1}{2009}\right)>4\)
=> A > 4
Ta có:\(\frac{2006}{2007}< 1\)
\(\frac{2007}{2008}< 1\)
\(\frac{2008}{2009}< 1\)
\(\frac{2009}{2006}>1\)\(\frac{2006}{2007}+\frac{2007}{2008}+\frac{2008}{2009}+\frac{2009}{2006}< 4\)
so sánh nha mn