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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

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 \(A>B\)

19 tháng 4 2017

Bài 1:

Ta có: \(A=\dfrac{2011+2012}{2012+2013}=\dfrac{2011}{2012+2013}+\dfrac{2012}{2012+2013}\)

Dễ thấy:

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

\(\Rightarrow A=\dfrac{2011}{2012+2013}+\dfrac{2012}{2012+2013}< B=\dfrac{2011}{2012}+\dfrac{2012}{2013}\)

Bài 2:

\(S=\dfrac{1}{4\cdot7}+\dfrac{1}{7\cdot10}+...+\dfrac{1}{37\cdot40}\)

\(=\dfrac{1}{3}\left(\dfrac{3}{4\cdot7}+\dfrac{3}{7\cdot10}+...+\dfrac{3}{37\cdot40}\right)\)

\(=\dfrac{1}{3}\left(\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+...+\dfrac{1}{37}-\dfrac{1}{40}\right)\)

\(=\dfrac{1}{3}\left(\dfrac{1}{4}-\dfrac{1}{40}\right)=\dfrac{1}{3}\cdot\dfrac{9}{40}=\dfrac{3}{40}< \dfrac{1}{3}\)

10 tháng 8 2017

Ta có : \(B=\dfrac{2011+2012}{2012+2013}=\dfrac{2011}{2012+2013}=\dfrac{2012}{2012+2013}\)

Mà : \(\dfrac{2011}{2012}>\dfrac{2011}{2012+2013}\)

\(\dfrac{2012}{2013}>\dfrac{2012}{2012+2013}\)

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

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

Vậy A > B