​khó thế

Ta có :M=\(\frac{2012^{37}+37^{2012}+1}{2012^{38}}\)=\(\frac{1}{2012}\)+\(\frac{37^{2012}}{2018^{38}}\)+\(\frac{1}{2012^{38}}\)

N=\(\frac{2012^{38}+37^{2012}+2}{2012^{39}}\)=\(\frac{1}{2012}\)+\(\frac{37^{2012}}{2012^{39}}\)+\(\frac{2}{2012^{39}}\)

Suy ra: M-N=\(\frac{37^{2012}}{2012^{38}}\left(1-\frac{1}{2012}\right)\)+\(\frac{1}{2012^{38}}\left(1-\frac{2}{2012}\right)\)

\(\Rightarrow\)M-N=\(\frac{37^{2012}}{2012^{38}}.\frac{2011}{2012}+\frac{1}{2012^{38}}.\frac{2010}{2012}\)

\(\Rightarrow\)M-N>0

\(\Rightarrow\)M>N

Vậy M>N

\(A=\frac{1}{2012}+\frac{37^{2012}}{2012^{38}}+\frac{1}{2012^{38}}\)

\(B=\frac{1}{2012}+\frac{37^{2012}}{2012^{39}}+\frac{2}{2012^{39}}\)

Ta có:

\(A-B=\frac{37^{2012}}{2012^{38}}-\frac{37^{2012}}{2012^{39}}+\frac{1}{2012^{38}}-\frac{2}{2012^{39}}\)

\(A-B=\frac{37^{2012}}{2012^{38}}\left(1-\frac{1}{2012}\right)+\frac{1}{2012^{38}}\left(1-\frac{2}{2012}\right)\)

\(A-B=\frac{37^{2012}}{2012^{38}}\left(\frac{2011}{2012}\right)+\frac{1}{2012^{38}}\left(\frac{2010}{2012}\right)\)

A - B > 0

=> A > B

A=2012³⁷/2012³⁸+ 37²⁰¹²/2012³⁸+1/2012³⁸

   = 1/2012+ 37²⁰¹²/2012³⁸+ 1/2012³⁸

Tương tự ta có: 

B=1/2012+ 37²⁰¹²/2012³⁹+1/2012³⁹+1/2012³⁹

Ta thấy: 1/2012=1/2012( ở 2 vế)

37²⁰¹²/2012³⁸ > 37²⁰¹²/ 2012³⁹( do cùng tử, mẫu nào nhỏ hơn thì phân số đó lớn hơn)

tương tự: 1/2012³⁸> 1/2012³⁹( 2012³⁸< 2012³⁹)

2012³⁹ là một số rất lớn nên 1/2012³⁹ rất bé và gần đến 0.

Vậy A>B.

A > B

tick nhé

Đặt \(A=\frac{37^{2013}+1}{37^{2012}+1}\) và \(B=\frac{37^{2014}+1}{37^{2013}+1}\) ta có : 

\(\frac{1}{37}A=\frac{37^{2013}+1}{37^{2013}+37}=\frac{37^{2013}+37-36}{37^{2013}+37}=\frac{37^{2013}+37}{37^{2013}+37}-\frac{36}{37^{2013}+37}=1-\frac{36}{37^{2013}+37}\)

\(\frac{1}{37}B=\frac{37^{2014}+1}{37^{2014}+37}=\frac{37^{2014}+37-36}{37^{2014}+37}=\frac{37^{2014}+37}{37^{2014}+37}-\frac{36}{37^{2014}+37}=1-\frac{36}{37^{2014}+37}\)

Vì \(\frac{36}{37^{2013}+37}>\frac{36}{37^{2014}+37}\) nên \(1-\frac{36}{37^{2013}+37}< 1-\frac{36}{37^{2014}+37}\)

\(\Rightarrow\)\(\frac{1}{37}A< \frac{1}{38}B\)

\(\Rightarrow\)\(A< B\)

Vậy \(A< B\)

Chúc bạn học tốt ~ 

\(A< B\)

Chúc bạn học tốt\(\approx\)

A = \(\frac{2011+2012}{2012+2013}=\frac{2011+2012}{4025}\)

Ta có:

\(\frac{2011}{2012}>\frac{2011}{4025}\)

\(\frac{2012}{2013}>\frac{2012}{4025}\)

=> \(\frac{2011}{2012}+\frac{2012}{2013}>\frac{2011}{4025}+\frac{2012}{4025}\)

=> \(B>\frac{2011+2012}{4025}\)

=>\(B>A\)

bạn xem ở đây nhé

Đây

A = B

mình giỏi lắm

IQ vô cực

?????????????????????????????????

 Cách 1

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

\(A=\frac{2011+1}{1+2013}\)

\(A=\frac{2012}{2014}\)

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

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

\(B=\frac{2011+1}{1+2013}\)

\(B=\frac{2012}{2014}\)

Vậy A và B bằng nhau vì cùng bằng \(\frac{2012}{2014}\)

Cách 2

A và B bằng nhau vì đều có hai phân số 2011/2012 + 2012/2013

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

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

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

Vậy A < B

Đặt a=1000^2012 thì \(A=\frac{a+2}{a-1}\)   ;   \(B=\frac{a}{a-3}\)

Xét    \(A-B=\frac{a+2}{a-1}-\frac{a}{a-3}=\frac{\left(a+2\right)\left(a-3\right)-a\left(a-1\right)}{\left(a-1\right)\left(a-3\right)}\)

                      \(=\frac{a^2-a-6-a^2+a}{\left(a-1\right)\left(a-3\right)}=\frac{-6}{\left(a-1\right)\left(a-3\right)}\)

Do \(a>1;a>3\)  nên \(\left(a-1\right)\left(a-3\right)>0\Leftrightarrow A-B< 0\)

 Do đó \(A>B\)