\(10A=\frac{2012^{2013}+10}{2012^{2013}+1}=\frac{2012^{2013}+1+9}{2012^{2013}+1}=1+\frac{9}{2012^{2013}+1}\)

\(10B=\frac{2012^{2012}+10}{2012^{2012}+1}=\frac{2012^{2012}+1+9}{2012^{2012}+1}=1+\frac{9}{2012^{2012}+1}\)

Vì \(\frac{9}{2012^{2013}+1}

ta co A=\(\frac{2012^{2012}+1}{2012^{2013}+1}< \frac{2012^{2012}+1+2011}{2012^{2013}+1+2011}\)=\(\frac{2012^{2012}+2012}{2012^{2013}+2012}=\frac{2012\left(2012^{2011}+1\right)}{2012\left(2012^{2012}+1\right)}\)

           =>A<B

Ta có : 

\(\frac{1}{2013}M=\frac{2013^{2012}+2012}{2013^{2012}+2013}=\frac{2013^{2012}+2013}{2013^{2012}+2013}-\frac{1}{2013^{2012}+2013}=1-\frac{1}{2013^{2012}+2013}\)

Lại có : 

\(\frac{1}{2013}N=\frac{2013^{2011}+2012}{2013^{2011}+2013}=\frac{2013^{2011}+2013}{2013^{2011}+2013}-\frac{1}{2013^{2011}+2013}=1-\frac{1}{2013^{2011}+2013}\)

Vì \(\frac{1}{2013^{2012}+2013}< \frac{1}{2013^{2011}+2013}\) nên \(M=1-\frac{1}{2013^{2012}}>N=1-\frac{1}{2013^{2011}+2013}\)

Vậy \(M>N\)

Chúc bạn học tốt ~ 

A<B bạn nhé

A=\(\frac{2012^{2012}+1}{2012^{2013}+1}\)

\(\Rightarrow\)A<\(\frac{2012^{2012}+1+2011}{2012^{2013}+1+2011}\)

           <\(\frac{2012^{2012}+2012}{2012^{2013}+2012}\)

           <\(\frac{2012\left(2012^{2011}+1\right)}{2012\left(2012^{2012}+1\right)}\)

           <\(\frac{2012^{2011}+1}{2012^{2012}+1}\)

          <B

Vậy A<B

A>B nhé

nhớ tink cho mình nhé!

A > B

231567

Ta có \(\frac{2012.2013}{2012.2013+1}\)và \(\frac{2013}{2012}\)

Vì \(\frac{2012.2013}{2012.2013+1}< 1< \frac{2013}{2012}\)

nên \(\frac{2012.2013}{2012.2013+1}< \frac{2013}{2012}\)

\(\frac{2012}{2013}\)và \(\frac{2011}{2012}\)

phàn bù của \(\frac{2012}{2013}\)là \(\frac{1}{2013}\)

phàn bù của \(\frac{2011}{2012}\)là \(\frac{1}{2012}\)

Vì \(\frac{1}{2012}>\frac{1}{2013}\Rightarrow\frac{2012}{2013}>\frac{2011}{2012}\)

Ta có : \(\frac{2012\cdot2013}{2012\cdot2013+1}< 1\)

             \(\frac{2013}{2012}>1\)

\(\Rightarrow\frac{2012\cdot2013}{2012\cdot2013+1}< \frac{2013}{2012}\)

Có : \(\frac{2012}{2013}=1-\frac{2012}{2013}=\frac{2013}{2013}-\frac{2012}{2013}=\frac{1}{2013}\)

         \(\frac{2011}{2012}=1-\frac{2011}{2012}=\frac{2012}{2012}-\frac{2011}{2012}=\frac{1}{2012}\)

Vì \(2013< 2012\)nên \(\frac{1}{2013}< \frac{1}{2012}\)hay \(\frac{2012}{2013}< \frac{2011}{2012}\)

So sánh các phân số sau :

\(\frac{2012\times2013}{2012\times2013+1}\)và \(\frac{2013}{2012}\)

Ta có : 

\(\frac{2012\times2013}{2012\times2013+1}\)

\(=\frac{1}{1}=1\)( cùng rút gọn 2012 x 2013 )

Vì \(1< \frac{2013}{2012}\)nên \(\frac{2012\times2013}{2012\times2013+1}< \frac{2013}{2013}\).

\(\frac{2012}{2013}\)và \(\frac{2011}{2012}\)

Ta có : 

\(1-\frac{2012}{2013}=\frac{1}{2013};1-\frac{2011}{2012}=\frac{1}{2012}\)

Vì \(\frac{1}{2013}< \frac{1}{2012}\)nên \(\frac{2012}{2013}< \frac{2011}{2012}\)

~ Chúc  bạn hok tốt ~

a) \(\frac{2^{10}+1}{2^{10}-1}\)và \(\frac{2^{10}-1}{2^{10}-3}\)

Ta có chính chất phân số trung gian là \(\frac{2^{10}+1}{2^{10}-3}\)

\(\frac{2^{10}+1}{2^{10}-1}>\frac{2^{10}+1}{2^{10}-3}\) ; \(\frac{2^{10}-1}{2^{10}-3}< \frac{2^{10}+1}{2^{10}-3}\)

Vì \(\frac{2^{10}+1}{2^{10}-1}>\frac{2^{10}+1}{2^{10}-3}>\frac{2^{10}-1}{2^{10}-3}\)

Nên \(\frac{2^{10}+1}{2^{10}-1}>\frac{2^{10}-1}{2^{10}-3}\)

b) \(A=\frac{2011}{2012}+\frac{2012}{2013}\)và \(B=\frac{2011+2012}{2012+2013}\)

Ta có : \(A=\frac{2011}{2012}+\frac{2012}{2013}>\frac{2011}{2013}+\frac{2012}{2013}=\frac{2011+2012}{2013}>\frac{2011+2012}{2012+2013}=B\)

Vậy A > B 

Có gì  sai cho sorry

a,

\(\frac{2^{10}+1}{2^{10}-1}=1+\frac{2}{2^{10}-1}< 1+\frac{2}{2^{10}-3}=\frac{2^{10}-1}{2^{10}-3}\)

b,

\(\frac{2011}{2012}+\frac{2012}{2013}>\frac{2011}{2012+2013}+\frac{2012}{2012+2013}=\frac{2011+2012}{2012+2013}\)

Ta có: \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2010^2}

Ta có \(B=\frac{2011}{2012}+\frac{2012}{2013}>\frac{2011}{2013}+\frac{2012}{2013}=\frac{2011+2012}{2013}\)

    Lại có: \(\frac{2011+2012}{2013}>\frac{2011+2012}{2012+2013}\)                        ( ngoặc 2 dòng này lại nhé dòng này và dòng trên)

 \(\Rightarrow B>A\)

nhầm là A > B