Question

Cho: \(A=\frac{11^{2007}+1}{11^{2008}+1}\) và \(B=\frac{11^{2008}+1}{11^{2009}+1}\)

Hãy so sánh A và B

Answer 1

Sửa lại:

Ta có: \(A=\frac{11^{2007}+1}{11^{2008}+1}\Rightarrow11A=\frac{11^{2008}+11}{11^{2008}+1}=1+\frac{10}{11^{2008}+1}\)

\(B=\frac{11^{2008}+1}{11^{2009}+1}\Rightarrow11B=\frac{11^{2009}+11}{11^{2009}+1}=1+\frac{10}{11^{2009}+1}\)

Vì \(\frac{10}{2^{2008}+1}>\frac{10}{11^{2009}+1}\Rightarrow1+\frac{10}{2^{2008}+1}>1+\frac{10}{11^{2009}+1}\)

\(\Rightarrow11A>11B\)

\(\Rightarrow A>B\)

Answer 2

Ta có: \(A=\frac{11^{2007}+1}{11^{2008}+1}\)

\(\Rightarrow11A=\frac{11^{2008}+11}{11^{2008}+1}=1+\frac{10}{11^{2008}+1}\)

\(B=\frac{11^{2008}+1}{11^{2009}+1}\)

\(\Rightarrow11B=\frac{11^{2009}+11}{11^{2009}+1}=1+\frac{10}{11^{2009}+1}\)

Vì \(\frac{10}{11^{2008}+1}< \frac{10}{11^{2009}+1}\Rightarrow1+\frac{10}{11^{2008}+1}< 1+\frac{10}{11^{2009}+1}\)

\(\Rightarrow11A< 11B\)

\(\Rightarrow A< B\)

Vậy \(A< B\)