\(A=\frac{100^{2007}+1}{100^{2009}+1}=>100^2A=\frac{100^2\left(100^{2007}+1\right)}{100^{2009}+1}\)\(=>100^2A=\frac{100^{2009}+100^2}{100^{2009}+1}=\frac{100^{2009}+1+9999}{100^{2009}+1}\)
\(=1+\frac{9999}{100^{2009}+1}\left(1\right)\)
\(B=\frac{100^{2005}+1}{100^{2007}+1}=>100^2B=\frac{100^2\left(100^{2005}+1\right)}{100^{2007}+1}\)\(=>100^2B=\frac{100^{2007}+100^2}{100^{2007}+1}=\frac{100^{2007}+1+9999}{100^{2007}+1}\)
\(=1+\frac{9999}{100^{2007}+1}\left(2\right)\)
Từ (1) và (2) ta thầy 1002009+1>1002007+1
=>\(\frac{9999}{100^{2009}+1}< \frac{9999}{100^{2007}+1}\)
\(=>1+\frac{9999}{100^{2009}+1}< 1+\frac{9999}{100^{2007}+1}\)=>1002A<1002B=>A<B
1002A=\(\frac{100^{2009}+100^2}{100^{2009}+1}=1+\frac{100^2-1}{100^{2009}+1}\)
1002B=\(\frac{100^{2007}+100^2}{100^{2007}+1}=1+\frac{100^2-1}{100^{2007}+1}\)
Vì \(\frac{100^2-1}{100^{2009}+1}< \frac{100^2-1}{100^{2007}+1}\) nên: 1002A<1002B
=>A<B
\(A=\frac{100^{2007}+1}{100^{2009}+1}=\frac{100^{2009}+1-100^2}{100^{2009}+1}=1-\frac{100^2}{100^{2009}+1}\)
\(B=\frac{100^{2005}+1}{100^{2007}+1}=\frac{100^{2007}+1-100^2}{100^{2007}+1}=1-\frac{100^2}{100^{2007}+1}\)
=> A>B