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17 tháng 5 2022

\(Q=\dfrac{2010+2011+2012}{2011+2012+2013}=\dfrac{2010}{2011+2012+2013}+\dfrac{2011}{2011+2012+2013}+\dfrac{2012}{2011+2012+2013}\)

Ta có: \(\dfrac{2010}{2011+2012+2013}< \dfrac{2010}{2011}\)

           \(\dfrac{2011}{2011+2012+2013}< \dfrac{2011}{2012}\)

           \(\dfrac{2012}{2011< 2012< 2013}< \dfrac{2012}{2013}\)

\(\Rightarrow\dfrac{2010}{2011+2012+2013}+\dfrac{2011}{2011+2012+2013}+\dfrac{2012}{2011+2012+2013}\)

\(\dfrac{2010}{2011}+\dfrac{2011}{2012}+\dfrac{2012}{2013}\)

\(P>Q\)

10 tháng 5 2017

\(Q=\dfrac{2010+2011+2012}{2011+2012+2013}=\dfrac{2010}{2011+2012+2013}+\dfrac{2011}{2011+2012+2013}+\dfrac{2012}{2011+2012+2013}\)Ta thấy:

\(\dfrac{2010}{2011}>\dfrac{2010}{2011+2012+2013}\\ \dfrac{2011}{2012}>\dfrac{2011}{2011+2012+2013}\\ \dfrac{2012}{2013}>\dfrac{2012}{2011+2012+2013}\\ \Rightarrow\dfrac{2010}{2011}+\dfrac{2011}{2012}+\dfrac{2012}{2013}>\dfrac{2010}{2011+2012+2013}+\dfrac{2011}{2011+2012+2013}+\dfrac{2012}{2011+2012+2013}\\ \Leftrightarrow P>Q\)

Vậy \(P>Q\)

2 tháng 4 2017

Ta có:

\(A=\dfrac{2010}{2011}+\dfrac{2011}{2012}\)

\(B=\dfrac{2010+2011}{2011+2012}\)

\(=\dfrac{2010}{2011+2012}+\dfrac{2011}{2011+2012}\)

Áp dụng tính chất \(\dfrac{a}{b}>\dfrac{a}{b+m}\) ta có:

\(\left\{{}\begin{matrix}\dfrac{2010}{2011}>\dfrac{2010}{2011+2012}\\\dfrac{2011}{2012}>\dfrac{2011}{2011+2012}\end{matrix}\right.\)

\(\Rightarrow\dfrac{2010}{2011}+\dfrac{2011}{2012}>\dfrac{2010}{2011+2012}+\dfrac{2011}{2011+2012}\)

Hay \(\dfrac{2010}{2011}+\dfrac{2011}{2012}>\dfrac{2010+2011}{2011+2012}\)

Vậy \(A>B\)

2.A=\(\dfrac{43.11}{2011^{2013}}\)+\(\dfrac{79}{2011^{2013}}\)=\(\dfrac{43.11+79}{2011^{2013}}\)

B=\(\dfrac{79.11}{2011^{2013}}\)+\(\dfrac{43}{2011^{2013}}\)=\(\dfrac{79.11+43}{2011^{2013}}\)

Ta có: 43.11+79=43.(10+1)+79=43.10+43+79=430+122

79.11+43=79.(10+1)+43=79.10+79+43=790+122

Vì 430+122<790+122 nên 43.11+79<79.11+43 (1)

Mà 20112013<20112013 (2)

Từ (1) và (2) suy ra A<B

3. A=\(\dfrac{2010.2012}{2011.2011}\)

Vì B<1 nên B>\(\dfrac{2010}{2012}\)=\(\dfrac{2010.2012}{2012.2012}\)

Vì 2010.2012=2010.2012; 2011.2011<2012.2012 nên B>A

4. A=\(\dfrac{3n}{3\left(2n+1\right)}\)=\(\dfrac{3n}{6n+3}\)

Vì 6n+3=6n+3; 3n<3n+1 nên A<B

\(2010A=\dfrac{2010^{2012}+2010}{2010^{2012}+1}=1+\dfrac{2009}{2010^{2012}+1}\)

\(2010B=\dfrac{2010^{2011}+2010}{2010^{2011}+1}=1+\dfrac{2009}{2010^{2011}+1}\)

mà \(2010^{2012}+1>2010^{2011}+1\)

nên A<B

9 tháng 2 2018

a/ Áp dụng bất đẳng thức :

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

Ta có :

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

\(\Leftrightarrow\dfrac{10^{2011}+1}{10^{2012}+1}< \dfrac{10^{2011}+1+9}{10^{2012}+1+9}=\dfrac{10^{2011}+10}{10^{2012}+10}=\dfrac{10\left(10^{2010}+1\right)}{10\left(10^{2011}+1\right)}=\dfrac{10^{2010}+1}{10^{2011}+1}\)

\(\Leftrightarrow\dfrac{10^{2011}+1}{10^{2012}+1}< \dfrac{10^{2010}+1}{10^{2011}+1}\)

9 tháng 2 2018

cho mn hỏi cái:\(\dfrac{10^{2011}+1}{10^{2012}+1}< \dfrac{10^{2011}+1+9}{10^{2012}+1+9}\)