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Ta có:4=1+1+1+1=\(\frac{2009}{2009}+\frac{2010}{2010}+\frac{2011}{2011}+\frac{2008}{2008}\)
\(\frac{2008}{2009}+\frac{1}{2009}+\frac{2009}{2010}+\frac{1}{2010}+\frac{2010}{2011}+\frac{1}{2011}+\frac{2008}{2008}\)
Xét \(A=\frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}+\frac{2011}{2008}\)
\(=\frac{2009}{2009}+\frac{2010}{2010}+\frac{2011}{2011}+\frac{2008}{2008}+\frac{1}{2008}+\frac{1}{2008}+\frac{1}{2008}\)
xét \(\frac{1}{2009}< \frac{1}{2008};\frac{1}{2010}< \frac{1}{2008};\frac{1}{2011}< \frac{1}{2008}\)
\(\Rightarrow4< A\)
\(B=\frac{2008+2009+2010}{2009+2010+2011}\)
\(=\frac{2008}{2009+2010+2011}+\frac{2009}{2009+2010+2011}+\frac{2010}{2009+2010+2011}\)
\(< \frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}=A\)
Bạn chỉ cần lấy : (2008/2009+2009/2010+2010/2011+2011/2008)-4=số dương
vậy (2008+...2008) > 4
Có 4 = 1+1+1+1
Vì 2008/2009<1 ; 2009/2010<1; 2010/2011<1;2011/2012<1
=>2008/2009+2009/2010 + 2010/2011+2011/2012<1+1+1+1=4
Do \(\frac{2008}{2009};\frac{2009}{2010};\frac{2010}{2011}\)bé hơn 1 , chỉ có phân số \(\frac{2011}{2008}\)lớn hơn 1 nên tổng của chúng bé hơn 4 .
\(\frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}+\frac{2011}{2008}=1-\frac{1}{2009}+1-\frac{1}{2010}+1-\frac{1}{2011}+1+\frac{3}{2008}=1+1+1+1+\frac{1}{2008}+\frac{1}{2008}+\frac{1}{2008}-\frac{1}{2009}-\frac{1}{2010}-\frac{1}{2011}=4+\left(\frac{1}{2008}-\frac{1}{2009}\right)+\left(\frac{1}{2008}-\frac{1}{2010}\right)+\left(\frac{1}{2008}-\frac{1}{2011}\right)\left(vì:2008>2009>2010>2011\right)\Rightarrow\frac{1}{2008}>\frac{1}{2009}>\frac{1}{2010}>\frac{1}{2011}\)
\(\Rightarrow\left\{{}\begin{matrix}\frac{1}{2008}-\frac{1}{2009}>0\\\frac{1}{2008}-\frac{1}{2010}>0\\\frac{1}{2008}-\frac{1}{2011}>0\end{matrix}\right.\Rightarrow4+\left(\frac{1}{2008}-\frac{1}{2009}\right)+\left(\frac{1}{2008}-\frac{1}{2010}\right)+\left(\frac{1}{2008}-\frac{1}{2011}\right)>4+0+0+0=4\Rightarrow\frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}+\frac{2011}{2008}>4\)
để mk chuyển dạng luôn nghe cho các bạn dễ làm
\(\frac{2008}{2004}+\frac{2009}{2010}+\frac{2010}{2011}+\frac{2011}{2008}\)
dạng tương tự là
\(\frac{28}{29}+\frac{29}{30}+\frac{30}{31}+\frac{31}{28}\)
\(mà\)\(vẫn\)\(so\)\(sánh\)\(với\)\(4\)
2008/2009 + 2009/2010 + 2010/2011 + 2011/2008
= 1 - 1/2009 + 1 - 1/2010 + 1 - 1/2011 + 1 + 3/2008
= (1 + 1 + 1 + 1) - (1/2009 + 1/2010 + 1/2011) + 3/2008
= 4 - (1/2009 + 1/2010 + 1/2011) + 3/2008
Vì 1/2009 < 1/2008
1/2010 < 1/2008
1/2011 < 1/2008
=> 1/2009 + 1/2010 + 1/2011 < 3 × 1/2008 = 3/2008
=> -(1/2009 + 1/2010 + 1/2011) + 3/2008 > 0
=> 4 - (1/2009 + 1/2010 + 1/2011) + 3/2008) > 4
=> 2008/2009 + 2009/2010 + 2010/2011 + 2011/2008 > 4