Vì \(\frac{10^{2011}+1}{10^{2012}+1}< 1\)

=> \(B=\frac{10^{2011}+1}{10^{2012}+1}< \frac{10^{2011}+1+9}{10^{2012}+1+9}=\frac{10^{2011}+10}{10^{2012}+10}=\frac{10\left(10^{2010}+1\right)}{10\left(10^{2011}+1\right)}=\frac{10^{2010}+1}{10^{2011}+1}=A\)

Vậy A > B

A>B hay sao y

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,

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

A < B nha!

\(B< \frac{10^{2012}+1+9}{10^{2013}+1+9}=\frac{10^{2012}+10}{10^{2013}+10}=\frac{10\left(10^{2011}+1\right)}{10\left(10^{2012}+1\right)}=\frac{10^{2011}+1}{10^{2012}+1}=A\)

Vậy A > B

Áp dụng bất đẳng thức :

\(\frac{a}{b}< 1\Leftrightarrow\frac{a}{b}< \frac{a+m}{b+m}\)

Ta có :

\(B=\frac{10^{2012}+1}{10^{2013}+1}< \frac{10^{2012}+1+9}{10^{2013}+1+9}=\frac{10^{2012}+10}{10^{2013}+10}=\frac{10\left(10^{2011}+1\right)}{10\left(10^{2012}+1\right)}=\frac{10^{2011}+1}{10^{2012}+1}=A\)

\(\Leftrightarrow B< A\)

B=\(\frac{2011^{10}-1}{2011^{10}-3}\) <1 => \(\frac{2011^{10}-1}{2011^{10}-3}\) < \(\frac{2011^{10}-1+2}{2011^{10}-3+2}\) = \(\frac{2011^{10}+1}{2011^{10}-1}\) = A

=> B<A

Cảm ơn bạn nhiều nha giải ra lại thấy dễ ak

b,Ta có 

\(\frac{2010}{2011}>\frac{2010}{2011+2012+2013}\)

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

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

\(\Rightarrow P>Q\)

\(A=\frac{-10}{20}+\frac{-10}{30}+\frac{-10}{42}+\frac{-10}{56}+\frac{-10}{72}+\frac{-10}{90}+\frac{-10}{110}\)

\(=-10\left(\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+\frac{1}{90}+\frac{1}{110}\right)\)

\(=-10\left(\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+\frac{1}{9.10}+\frac{1}{10.11}\right)\)

\(=-10\left(\frac{1}{4}-\frac{1}{11}\right)\)

\(=\frac{-35}{22}\)

Có \(\hept{\begin{cases}A=\frac{-9}{10^{2012}}+\frac{-19}{10^{2011}}\\B=\frac{-19}{10^{2012}}+\frac{-9}{10^{2011}}\end{cases}}\)

\(\Rightarrow\)A-B=\(\frac{10}{10^{2011}}-\frac{10}{10^{2012}}=\frac{1}{10^{2010}}-\frac{1}{10^{2011}}>0\)

\(\Rightarrow A>B\)

a) Ta có :

\(A=\frac{10^{2010}+1}{10^{2011}+1}\)

\(\Rightarrow10A=\frac{10^{2011}+10}{10^{2011}+1}=\frac{\left(10^{2011}+1\right)+9}{10^{2011}+1}=1+\frac{9}{10^{2011}+1}\)

\(B=\frac{10^{2011}+1}{10^{2012}+1}\)

\(\Rightarrow10B=\frac{10^{2012}+10}{10^{2012}+1}=\frac{\left(10^{2012}+1\right)+9}{10^{2012}+1}=1+\frac{9}{10^{2012}+1}\)

Vì \(\frac{9}{10^{2011}+1}>\frac{9}{10^{2012}+1}\)nên \(10A>10B\)

\(\Rightarrow A>B\)

Vậy : \(A>B\)

b) Ta có :

\(\left(\frac{-1}{2}\right)^{11}=\frac{-1^{11}}{2^{11}}=\frac{-1}{2^{11}}\)

\(\left(\frac{-1}{2}\right)^{13}=\frac{-1^{13}}{2^{13}}=\frac{-1}{2^{13}}\)

Vì \(\frac{-1}{2^{11}}>\frac{-1}{2^{13}}\)nên \(\left(\frac{-1}{2}\right)^{11}>\left(\frac{-1}{2}\right)^{13}\)

Vậy : \(\left(\frac{-1}{2}\right)^{11}>\left(\frac{-1}{2}\right)^{13}\)

\(B=\frac{10^{2011}+1}{10^{2012}+1}< \frac{10^{2011}+1+9}{10^{2012}+1+9}\)

\(B=\frac{10^{2011}+1}{10^{2012}+1}< \frac{10^{2011}+10}{10^{2012}+10}\)

\(B=\frac{10^{2011}+1}{10^{2012}+1}< \frac{10\cdot\left(10^{2010}+1\right)}{10\cdot\left(10^{2011}+1\right)}=\frac{10^{2010}+1}{10^{2011}+1}=A\)

Vậy : B < A

So sánh 2 phân số sau  $\frac{10^{2011}+10}{10^{2012}+10}v\text{à}\frac{10^{2012}-10}{10^{2013}-10}$102011+10102012+10 và102012−10102013−10 

kick dzô chữ xanh là được!! OK

Ta có : 

10. A = \(\frac{10.\left(10^{2011}+1\right)}{10^{2012}+1}\)

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

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

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

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

10 . B = \(\frac{10.\left(10^{2012}+1\right)}{10^{2013}+1}\)

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

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

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

Vì \(\frac{9}{10^{2012}+1}\) >\(\frac{9}{10^{2013}+1}\)  nên 10.A > 10.B

=> A >B 

Vậy ...........