So sánh: (2^2010+1)/(2^2017+1) và (2^2012+1)/(2^2009+1)
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Ta có :
\(N=\frac{2009^{2010}-2}{2009^{2011}-2}< \frac{2009^{2010}-2+2011}{2009^{2011}-2+2011}\)
\(=\frac{2009^{2010}+2009}{2009^{2011}+2009}=\frac{2009.\left(2009^{2009}+1\right)}{2009.\left(2009^{2010}+1\right)}\)
\(=\frac{2009^{2009}+1}{2009^{2010}+1}=M\)
Vậy \(M>N\)
Ta có: \(B< 1\)
\(\Rightarrow B< \frac{2009^{2010}-2+3}{2009^{2011}-2+3}=\frac{2009^{2010}+1}{2009^{2011}+1}\left(1\right)\)
Mà \(\frac{2009^{2010}+1}{2009^{2011}+1}< 1\)
\(\Rightarrow\frac{2009^{2010}+1}{2009^{2011}+1}< \frac{2009^{2010}+1+2008}{2009^{2011}+1+2008}=\frac{2009^{2010}+2009}{2009^{2011}+2009}=\frac{2009\left(2009^{2009}+1\right)}{2009\left(2009^{2010}+1\right)}=\frac{2009^{2009}+1}{2009^{2010}+1}=A\left(2\right)\)
Từ (1) và (2) suy ra A > B
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Ta có :
\(B=\dfrac{2009^{2010}-2}{2009^{2011}-2}< 1\)
\(\Leftrightarrow B< \dfrac{2009^{2010}-2+2011}{2009^{2011}-2+2011}=\dfrac{2009^{2010}+2009}{2009^{2011}+2009}=\dfrac{2009\left(2009^{2009}+1\right)}{2009\left(2009^{2010}+1\right)}=\dfrac{2009^{2009}+1}{2009^{2010}+1}=A\)
\(\Leftrightarrow A>B\)
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(2 ^2010 +1)/(2 ^2017 +1) và (2 ^2012 +1)/(2 ^2009 +1)
Trả lời :
(2 ^2010 +1)/(2 ^2017 +1) < (2 ^2012 +1)/(2 ^2009 +1)
HC T bài này khó đó
ta có:
\(\left(2^{2010}+1\right)>\left(2^{2009}+1\right)>1\) và \(\left(2^{2017}+1\right)>\left(2^{2012}+1\right)>1\)
thế nên
\(\left(2^{2010}+1\right)\left(2^{2017}+1\right)>\left(2^{2012}+1\right)\left(2^{2009}+1\right)\)