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Ta có : Phân số trung gian của hai phân số là \(\frac{-2011}{1931}\)
So sánh : \(\frac{-2011}{1931}\)>\(\frac{-2011}{2038}\); \(\frac{-2011}{1931}\)<\(\frac{-1904}{1931}\)
=>\(\frac{-2011}{2038}\)<\(\frac{-1904}{1931}\)
Ta có :
\(\frac{-2011}{2038}>\frac{-1904}{2038}>\frac{-1904}{1931}\)
Vậy \(\frac{-2011}{2038}>\frac{-1904}{1931}\)
ta thấy
\(\frac{2011}{2038}+\frac{27}{2038}=1\)
\(\frac{1904}{1931}+\frac{27}{1931}=1\)
mà\(\frac{27}{2038}>\frac{27}{1931}\Rightarrow\frac{2011}{2038}< \frac{1904}{1931}\Rightarrow\frac{-2011}{2038}>\frac{-1904}{1931}\)
vậy...
Ta có: B=\(\frac{17^{2009}+1}{17^{2010}+1}\)<1 ( Vì 172009+1< 172010+1 )
Nên B=\(\frac{17^{2009}+1}{17^{2010}+1}\)<\(\frac{17^{2009}+1+16}{17^{2010}+1+16}\)
=\(\frac{17^{2009}+17}{17^{2010}+17}\)
=\(\frac{17\left(17^{2008}+1\right)}{17\left(17^{2009}+1\right)}\)
=\(\frac{17^{2008+1}}{17^{2009}+1}\)=A
Vậy A>B
\(A=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2012}}{1+\frac{2012}{2011}+\frac{2012}{2010}+\frac{2012}{2009}+...+\frac{2012}{2}}\)
\(=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2012}}{\frac{2012}{2012}+\frac{2012}{2011}+...+\frac{2012}{2}}\)
\(=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2012}}{2012\left(\frac{1}{2012}+\frac{1}{2011}+...+\frac{1}{2}\right)}=\frac{1}{2012}\)
`-2011/2038 = -1 + 27/2038`
`-1904/1931 = -1 + 27/1931`.
Vì `27/1931 > 27/2038`.
`=> -2011/2038 < -1904/1931`.
\(\dfrac{-2011}{2038}< \dfrac{-1904}{1931}\)