2008/2009 - 2009/2008 + 1/2009+ 2007/2008
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Đặt \(A=\frac{2009^{2008}-1}{2009^{2009}-1}\)
\(\Rightarrow2009A=\frac{2009.\left(2009^{2008}-1\right)}{2009^{2009}-1}=\frac{2009^{2009}-2009}{2009^{2009}-1}\)
\(=\frac{2009^{2009}-1-2008}{2009^{2009}-1}=1-\frac{2008}{2009^{2009}-1}\)
Đặt \(B=\frac{2009^{2007}+1}{2009^{2008}+1}\)
\(\Rightarrow2009B=\frac{2009.\left(2009^{2007}+1\right)}{2009^{2008}+1}=\frac{2009^{2008}+2009}{2009^{2008}+1}\)
\(=\frac{2009^{2008}+1+2008}{2009^{2008}+1}=1+\frac{2008}{2009^{2008}+1}\)
Vì : \(\frac{2008}{2009^{2009}-1}< \frac{2008}{2009^{2008}+1}\)
\(\Rightarrow A=1-\frac{2008}{2009^{2009}-1}< B=1+\frac{2008}{2009^{2008}+1}\)
Vậy \(\frac{2009^{2008}-1}{2009^{2009}-1}< \frac{2009^{2007}+1}{2009^{2008}+1}\)
\(b,S=\frac{2007}{2008}+\frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}\)
\(\text{Ta có: }\frac{2007}{2008}< 1\)
\(\frac{2008}{2009}< 1\)
\(\frac{2009}{2010}< 1\)
\(\frac{2010}{2011}< 1\)
\(\Rightarrow\frac{2007}{2008}+\frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}< 1+1+1+1\)
\(\Rightarrow\frac{2007}{2008}+\frac{2008}{2009}+\frac{2009}{2010}+\frac{2010}{2011}< 4\)
So sánh A và B, biết :
A= 2008 / 977654321 + 2009 / 246813579
B= 2009 / 987654321 + 2008 / 246813579
A < B = B > A
*Ryeo*
\(A=\frac{2008}{977654321}+\frac{2008}{246813579}+\frac{1}{246813579}\)
\(B=\frac{2009}{987654321}+\frac{2008}{246813579}\)
Thấy \(\frac{2008}{977654321}=2008\cdot\frac{1}{977654321}\)với \(\frac{1}{977654321}>\frac{1}{987654321}\)và\(2008>\frac{1}{987654321}\)nên \(\frac{2008}{977654321}>\frac{1}{987654321}\)
Ta cũng có \(\frac{1}{246813579}>\frac{1}{987654321}\)và \(\frac{2008}{246813579}=\frac{2008}{246813579}\)nên A > B.
Vậy A > B
8/8=1
\(\frac{2008}{2009}\) \(-\frac{2009}{2008}\) \(+\frac{1}{2009}\) \(+\frac{2007}{2008}\)
\(=\frac{2009}{2008}\) \(+\frac{2007}{2008}\) \(-\frac{2008}{2009}\) \(+\frac{1}{2009}\)
\(=2-1\)
\(=1\)
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