Question

 So sánh : A= 10^2006+1/10^2007+1 ; B= 10^2007+1/10^2008+1

Answer 1

Ta có: A=\(\frac{10^{2006}+1}{10^{2007}+1}\)

=>10A=\(\frac{10\left(10^{2006}+1\right)}{10^{2007}+1}=\frac{10^{2007}+10}{10^{2007}+1}=1+\frac{9}{10^{2007}+1}\)             

Ta có: B=\(\frac{10^{2007}+1}{10^{2008}+1}\)

=>10B=\(\frac{10\left(10^{2007}+1\right)}{10^{2008}+1}=\frac{10^{2008}+10}{10^{2008}+1}=1+\frac{9}{10^{2008}+1}\)  

Mà \(\frac{9}{10^{2007}+1}>\frac{9}{10^{2008}+1}\)        (do 10²⁰⁰⁷+1<10²⁰⁰⁸+1)

=>\(1+\frac{9}{10^{2007}+1}>1+\frac{9}{10^{2008}+1}\)

=>10A>10B

=>A>B

Answer 2

Áp dụng a/b < 1 => a/b < a+m/b+m (a,b,m thuộc N*)

=> \(B=\frac{10^{2007}+1}{10^{2008}+1}< \frac{10^{2007}+1+9}{10^{2008}+1+9}\)

=> \(B< \frac{10^{2007}+10}{10^{2008}+10}\)

=> \(B< \frac{10.\left(10^{2006}+1\right)}{10.\left(10^{2007}+1\right)}\)

=> \(B< \frac{10^{2006}+1}{10^{2007}+1}=A\)