ý, nếu không được dùng cách kia thì làm cách này cho chắc đi :v

Ta có: \(2008A=\frac{2008\left(2008^{2008}+1\right)}{2008^{2009}+1}=\frac{2008^{2009}+2008}{2008^{2009}+1}=\frac{\left(2008^{2009}+1\right)+2007}{2008^{2009}+1}=1+\frac{2007}{2008^{2009}+1}\)

Lại có: \(2008B=\frac{2008\left(2008^{2007}+1\right)}{2008^{2008}+1}=\frac{2008^{2008}+2008}{2008^{2008}+1}=\frac{\left(2008^{2008}+1\right)+2007}{2008^{2008}+1}=1+\frac{2007}{2008^{2008}+1}\)

Vì 2008 < 2009 \(\Rightarrow2008^{2008}< 2008^{2009}\)\(\Rightarrow2008^{2008}+1< 2008^{2009}+1\)\(\Rightarrow\frac{2007}{2008^{2008}+1}>\frac{2007}{2008^{2009}+1}\)\(\Rightarrow1+\frac{2007}{2008^{2008}+1}>1+\frac{2007}{2008^{2009}+1}\)\(\Rightarrow2008B>2008A\)\(\Rightarrow B>A\)

Vì A <1 , B < 1

Nên ta có: \(A=\frac{2008^{2008}+1}{2008^{2009}+1}< \frac{2008^{2008}+1+2007}{2008^{2009}+1+2007}=\frac{2008^{2008}+2008}{2008^{2009}+2008}=\frac{2008\left(2008^{2007}+1\right)}{2008\left(2008^{2008}+1\right)}=\frac{2008^{2007}+1}{2008^{2008}+1}=B\)

A=\(\frac{2007^{2007}}{2008^{2008}}\)

B=\(\frac{2008^{2008}}{2009^{2009}}\)

A be honB

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a) Áp dụng \(\frac{a}{b}< 1\Leftrightarrow\frac{a}{b}< \frac{a+m}{b+m}\) (a;b;m \(\in\) N*)

Ta có:

\(A=\frac{2008^{2008}+1}{2008^{2009}+1}< \frac{2008^{2008}+1+2007}{2009^{2009}+1+2007}\)

\(A< \frac{2008^{2008}+2008}{2008^{2009}+2008}\)

\(A< \frac{2008.\left(2008^{2007}+1\right)}{2008.\left(2008^{2008}+1\right)}=\frac{2008^{2007}+1}{2008^{2008}+1}=B\)

=> A < B

b) Áp dụng \(\frac{a}{b}>1\Leftrightarrow\frac{a}{b}>\frac{a+m}{b+m}\) (a;b;m \(\in\) N*)

Ta có: 

\(N=\frac{100^{101}+1}{100^{100}+1}>\frac{100^{101}+1+99}{100^{100}+1+99}\)

\(N>\frac{100^{101}+100}{100^{100}+100}\)

\(N>\frac{100.\left(100^{100}+1\right)}{100.\left(100^{99}+1\right)}=\frac{100^{100}+1}{100^{99}+1}=M\)

=> M > N

\(A=\dfrac{2008^{2008}+1}{2008^{2009}+1}\)

\(2008\cdot A=\dfrac{2008^{2009}+2008}{2008^{2009}+1}\)

\(=\dfrac{2008^{2009}+1+2007}{2008^{2009}+1}\)

\(=1+\dfrac{2007}{2008^{2009}+1}\)

\(B=\dfrac{2008^{2007}+1}{2008^{2008}+1}\)

\(2008\cdot B=\dfrac{2008^{2008}+2008}{2008^{2008}+1}\)

\(=\dfrac{2008^{2008}+1+2007}{2008^{2008}+1}\)

\(=1+\dfrac{2007}{2008^{2008}+1}\)

Ta có: \(2008^{2009}+1>2008^{2008}+1\)

\(\Rightarrow\dfrac{1}{2008^{2009}+1}< \dfrac{1}{2008^{2008}+1}\)

\(\Rightarrow\dfrac{2007}{2008^{2009}+1}< \dfrac{2007}{2008^{2008}+1}\)

\(\Rightarrow1+\dfrac{2007}{2008^{2009}+1}< 1+\dfrac{2007}{2008^{2008}+1}\)

hay \(A < B\)

#\(Toru\)

Bài 1:

Ta có: 2009²⁰=(2009²)¹⁰=4036081¹⁰ ;  20092009¹⁰=20092009¹⁰

Vì 4036081¹⁰< 20092009¹⁰ => 2009²⁰< 20092009¹⁰

Bài 1:

Ta có:\(2009^{20}\)=\(2009^{10}\).\(2009^{10}\)

         \(20092009^{10}\)=(\(\left(2009.10001\right)^{10}=2009^{10}.10001^{10}\)

Vì 2009<10001\(\Rightarrow2009^{20}< 20092009^{10}\)