Question

so sánh

2011^2010+1/2011^2011+1     và 2011^2011+1/2011^2012+1

Answer 1

Đặt \(A=\frac{2011^{2010}+1}{2011^{2011}+1}\Rightarrow2011A=\frac{2011^{2011}+2011}{2011^{2011}+1}=1+\frac{2010}{2011^{2011}+1}\)

\(B=\frac{2011^{2011}+1}{2011^{2012}+1}\Rightarrow2011B=\frac{2011^{2012}+2011}{2011^{2012}+1}=1+\frac{2010}{2011^{2012}+1}\)

\(2011^{2011}+1< 2011^{2012}+1\)

\(\Rightarrow\frac{2010}{2011^{2011}+1}>\frac{2010}{2011^{2012}+1}\)

\(\Rightarrow2011A>2011B\Rightarrow A>B\)

\(\Rightarrow\frac{2011^{2010}+1}{2011^{2011}+1}>\frac{2011^{2011}+1}{2011^{2012}+1}\)