​khó thế

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

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

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

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

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) Ta có:

5³⁷ > 5³⁶ = (5³)¹² = 125¹²

11²⁴ = (11²)¹² = 121¹²

Vì 5³⁷ > 125¹² > 121¹²

=> 5³⁷ > 11²⁴

b) + Nếu a < b

=> 2012a < 2012b

=> 2012a + ab < 2012b + ab

=> a.(b + 2012) < b.(a + 2012)

=> \(\frac{a}{b}< \frac{a+2012}{b+2012}\)

+ Nếu a = b

=> 2012a = 2012b

=> 2012a + ab = 2012b + ab

=> a.(b + 2012) = b.(a + 2012)

=> \(\frac{a}{b}=\frac{a+2012}{b+2012}\)

+ Nếu a > b

=> 2012a > 2012b

=> 2012a + ab > 2012b + ab

=> a.(b + 2012) > b.(a + 2012)

=> \(\frac{a}{b}>\frac{a+2012}{b+2012}\)

ÁP DỤNG CÔNG THỨC NẾU \(\frac{a}{b}\)>1 thì

\(\frac{a}{b}\)>\(\frac{a+m}{b+m}\)

Ta có : \(\frac{2012^{12}+1}{2012^{13}+1}\)>\(\frac{2012^{12}+1+2011}{2012^{13}+1+2011}\)=\(\frac{2012^{12}+2012}{2012^{13}+2012}\)=\(\frac{2012.\left(2012^{11}+1\right)}{2012.\left(2012^{12}+1\right)}\)

rồi rút gọn thành \(\frac{2012^{11}+1}{2012^{12}+1}=B\)

Vậy A>B

Nhớ cho mình đúng nha

Ta có:\(A=\dfrac{2012^{2012}+1}{2012^{2013}+1}\)

\(\Rightarrow2012.A=\dfrac{2012^{2013}+2012}{2012^{2013}+1}=\dfrac{2012^{2013}+1+2011}{2012^{2013}+1}=1+\dfrac{2011}{2012^{2013}+1}\)Ta có:\(B=\dfrac{2012^{2011}+1}{2012^{2012}+1}\)

\(\Rightarrow2012.B=\dfrac{2012^{2012}+2012}{2012^{2012}+1}=\dfrac{2012^{2012}+1+2011}{2012^{2012}+1}=1+\dfrac{2011}{2012^{2012}+1}\)Vì\(\dfrac{2011}{2012^{2013}+1}< \dfrac{2011}{2012^{2012}+1}\)

\(\Rightarrow1+\dfrac{2011}{2012^{2013}+1}< 1+\dfrac{2011}{2012^{2012}+1}\)

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

Vậy A<B

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

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

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

a, Ta có : \(7x+4y⋮37\)

\(\Rightarrow23\left(7x+4y\right)⋮37\)

\(\Rightarrow161x+92y⋮37\)

\(\Rightarrow\left(13x+18y\right)+148x+74y⋮37\)

Mà \(\hept{\begin{cases}148x⋮37\\74x⋮37\end{cases}\Rightarrow13x+18y⋮37}\)

Vậy \(13x+18y⋮37\)

b, Ta có : \(A=\frac{2014^{2012}+1}{2014^{2013}+1}\)

\(\Rightarrow2014A=\frac{2014^{2013}+2014}{2014^{2013}+1}=\frac{2014^{2013}+1+2013}{2014^{2013}+1}=1+\frac{2013}{2014^{2013}+1}\)

Ta có : \(B=\frac{2014^{2011}+1}{2014^{2012}+1}\)

\(\Rightarrow2014B=\frac{2014^{2012}+2014}{2014^{2012}+1}=\frac{2014^{2012}+1+2013}{2014^{2012}+1}=1+\frac{2013}{2014^{2012}+1}\)

Vì \(2014^{2013}+1>2014^{2012}+1\)

\(\Rightarrow\frac{1}{2014^{2013}+1}< \frac{1}{2014^{2012}+1}\Rightarrow1+\frac{1}{2014^{2013}+1}< 1+\frac{1}{2014^{2012}+1}\)

\(\Rightarrow2014A< 2014B\Rightarrow A< B\)