Question

so sánh:

A=\(\frac{5^{2010}+1}{5^{2011}+1}\)và B=\(\frac{5^{2009}+1}{5^{2010}+1}\)

Answer 1

Ta có: \(5A=\frac{5^{2011}+5}{5^{2011}+1}=\frac{5^{2011}+1+4}{5^{2011}+1}=1+\frac{4}{5^{2011}+16}\)

\(5B=\frac{5^{2010}+5}{5^{2010}+1}=\frac{5^{2010}+1+4}{5^{2010}+1}=1+\frac{4}{5^{2010}+1}\)

Vì \(\frac{4}{5^{2011}+1}< \frac{4}{5^{2010}+1}\Rightarrow5A< 5B\Rightarrow A< B\)

Answer 2

Ta có:

A = \(\frac{5^{2010}+1}{5^{2011}+1}\)

\(\Rightarrow5A=\frac{5.\left(5^{2010}+1\right)}{5^{2011}+1}\)\(=\frac{5^{2011}+5}{5^{2011}+1}=1+\frac{4}{5^{2011}+1}\)

B=\(\frac{5^{2009}+1}{5^{2010}+1}\)

\(\Rightarrow5B=\frac{5.\left(5^{2009}+1\right)}{5^{2010}+1}=\frac{5^{2010}+5}{5^{2010}+1}=1+\frac{4}{5^{2010}+1}\)

Ta thấy \(5^{2011}+1>5^{2010}+1\)

\(\Rightarrow\frac{4}{5^{2011}+1}< \frac{4}{5^{2010}+1}\)

\(\Rightarrow1+\frac{4}{5^{2011}+1}< 1+\frac{4}{5^{2010}+1}\)

Hay 5.A<5.B

Vậy A<B (đpcm)