\(10A=\dfrac{10^{2007}+10}{10^{2007}+1}=\dfrac{10^{2007}+1+9}{10^{2007}+1}=1+\dfrac{9}{10^{2007}+1}\left(1\right)\)\(10B=\dfrac{10^{2008}+10}{10^{2008}+1}=\dfrac{10^{2008}+1+9}{10^{2008}+1}=1+\dfrac{9}{10^{2008}+1}\left(2\right)\)Từ (1) và ( 2 ) suy ra A>B
Cách 2 :
Ta CM BĐT sau :
\(\dfrac{a}{b}< \dfrac{a+m}{b+m}\left(a< b;a;b;m>0\right)\)
Ta có :
\(a< b\\ \Rightarrow am< bm\\ \Rightarrow ab+am< bm+ab\\ \Rightarrow a\left(b+m\right)< b\left(a+m\right)\\ \Rightarrow\dfrac{a}{b}< \dfrac{a+m}{b+m}\)
\(\Rightarrow A=\dfrac{10^{2007}+1}{10^{2008}+1}< \dfrac{10^{2007}+1+9}{10^{2008}+1+9}\\ =\dfrac{10\left(10^{2006}+1\right)}{10\left(10^{2007}+1\right)}=\dfrac{10^{2006}+1}{10^{2007}+1}=B\\ \Rightarrow A< B\)
