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\(A=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2012}}{1+\frac{2012}{2011}+\frac{2012}{2010}+\frac{2012}{2009}+...+\frac{2012}{2}}\)
\(=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2012}}{\frac{2012}{2012}+\frac{2012}{2011}+...+\frac{2012}{2}}\)
\(=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2012}}{2012\left(\frac{1}{2012}+\frac{1}{2011}+...+\frac{1}{2}\right)}=\frac{1}{2012}\)
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\(C=\left(1+\frac{2}{3}\right)\cdot\left(1+\frac{2}{5}\right)\cdot\left(1+\frac{2}{7}\right)\cdot\cdot\cdot\cdot\cdot\left(1+\frac{2}{2009}\right)\cdot\left(1+\frac{2}{2011}\right)\)
\(C=\frac{5}{3}\cdot\frac{7}{5}\cdot\frac{9}{7}\cdot\cdot\cdot\cdot\cdot\frac{2011}{2009}\cdot\frac{2013}{2011}\)
\(C=\frac{5\cdot7\cdot9\cdot\cdot\cdot\cdot\cdot2011\cdot2013}{3\cdot5\cdot7\cdot\cdot\cdot\cdot\cdot2009\cdot2011}\)
\(C=\frac{2013}{3}\)
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C=(1+2/3).(1+2/5).(1+2/7)......(1+2/2009).(1+2/2011)
C=5/3.7/5.9/7......2011/2009.2013/2011
C=5.7.9.....2013/3.5.7.....2009.2011
C=2013/3
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1)Đặt A=1+2+22+23+.....+22008
=>2A=2+22+23+....+22009
=>2A-A=(2+22+23+...+22009)-(1+2+22+23+....+22008)
=-1+22009
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a,Ta co:\(A=\frac{2005^{2005}+1}{2005^{2006}+1}<\frac{2005^{2005}+1+2004}{2005^{2006}+1+2004}=\frac{2005^{2005}+2005}{2005^{2006}+2005}\)
\(=\frac{2005\left(2005^{2004}+1\right)}{2005\left(2005^{2005}+1\right)}=\frac{2005^{2004}+1}{2005^{2005}+1}\) =B Vay A<B
b,lam tuong tu nhu y a
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C=5/3×7/5×9/7×...2011/2009×2013/2011
C=5×7×9×...×2011×2013/3×5×7×...×2009×2011
C=2013/3
C=674
Nếu sai bảo mình nhé !
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\(C = \frac{5}{3} . \frac{7}{5} . ... \frac{2013}{2011}\)
\(C = \frac{5.7.9....2011.2013}{3.5.7...2011}\)
\(C = \frac{5.2013}{3} = \frac{10065}{3} = 3355\)