$\frac{\frac{2010}{2011}}{\frac{2012}{2013}}+\frac{\frac{2011}{2012}}{\frac{2013}{2014}}+\frac{\frac{2012}{2013}}{\frac{2014}{2015}}$

$\frac{\frac{2010}{2011}}{\frac{2012}{2013}}+\frac{\frac{2011}{2012}}{\frac{2013}{2014}}+\frac{\frac{2012}{2013}}{\frac{2014}{2015}}$

$\frac{\frac{2010+2011+2012}{2011+2012+2013}}{\frac{2012+2013+2014}{2013+2014+2015}}$

$\frac{\frac{2010}{2011}+\frac{2011}{2012}+\frac{2012}{2013}}{\frac{2012+2013+2014}{2013+2014+2015}}$

$\frac{\frac{2010+2011+2012}{2011+2012+2013}}{\frac{2012}{2013}+\frac{2013}{2014}+\frac{2014}{2015}}$

dễ ợt nhưng éo biết làm thông cảm nha

ban Dang Ha Trang an noi gi ki vay 

\(\frac{2014}{2013}+\frac{2013}{2012}+\frac{2012}{2011}+\frac{2011}{2014}\)

\(=1+\frac{1}{2013}+1+\frac{1}{2012}+1+\frac{1}{2011}+1-\frac{3}{2014}\)

\(=4+\left(\frac{1}{2013}+\frac{1}{2012}+\frac{1}{2011}-\frac{1}{2014}-\frac{1}{2014}-\frac{1}{2014}\right)\)

Ta có:

 \(\frac{1}{2011}>\frac{1}{2014}\Rightarrow\frac{1}{2011}-\frac{1}{2014}>0\)

\(\frac{1}{2012}>\frac{1}{2014}\Rightarrow\frac{1}{2012}-\frac{1}{2014}>0\)

\(\frac{1}{2013}>\frac{1}{2014}\Rightarrow\frac{1}{2013}-\frac{1}{2014}>0\)

\(\Rightarrow\frac{1}{2011}-\frac{1}{2014}+\frac{1}{2012}-\frac{1}{2014}+\frac{1}{2013}-\frac{1}{2014}>0\)

\(\Rightarrow4+\left(\frac{1}{2013}+\frac{1}{2012}+\frac{1}{2011}-\frac{1}{2014}-\frac{1}{2014}-\frac{1}{2014}\right)>4\)( thêm 2 vế với 4 )

\(\Rightarrow\frac{2014}{2013}+\frac{2013}{2012}+\frac{2012}{2011}+\frac{2011}{2014}>4\)

Vậy \(\frac{2014}{2013}+\frac{2013}{2012}+\frac{2012}{2011}+\frac{2011}{2014}>4\) 

Tham khảo nhé~

Mỗi số hạng của tổng đều nhỏ hơn 1 => Tổng đó nhỏ hơn 4

Ta có:

\(\frac{2014}{2013}+\frac{2013}{2012}+\frac{2012}{2011}+\frac{2011}{2014}=4+\frac{1}{2013}+\frac{1}{2012}+\frac{1}{2011}-\frac{3}{2014}\)

Vì\(\frac{1}{2013}>\frac{1}{2014},\frac{1}{2012}>\frac{1}{2014},\frac{1}{2011}>\frac{1}{2014}\)

=>\(\frac{1}{2013}+\frac{1}{2012}+\frac{1}{2011}>\frac{3}{2014}\)

=>\(\frac{1}{2013}+\frac{1}{2012}+\frac{1}{2011}-\frac{3}{2014}>0\)

=>\(4+\frac{1}{2013}+\frac{1}{2012}+\frac{1}{2011}-\frac{3}{2014}>4\)

mỗi  số hạng trong biểu thức A đều nhỏ hơn 1 mà có 15 số nên tổng A sẽ nhỏ hơn 15

ta thay tong tren <1+1+1+1+1+1+1+1+1+1+1+1+1+1+1

hay tong tren be hon 15

Ta có \(\frac{2012.2013}{2012.2013+1}\)và \(\frac{2013}{2012}\)

Vì \(\frac{2012.2013}{2012.2013+1}< 1< \frac{2013}{2012}\)

nên \(\frac{2012.2013}{2012.2013+1}< \frac{2013}{2012}\)

\(\frac{2012}{2013}\)và \(\frac{2011}{2012}\)

phàn bù của \(\frac{2012}{2013}\)là \(\frac{1}{2013}\)

phàn bù của \(\frac{2011}{2012}\)là \(\frac{1}{2012}\)

Vì \(\frac{1}{2012}>\frac{1}{2013}\Rightarrow\frac{2012}{2013}>\frac{2011}{2012}\)

Ta có : \(\frac{2012\cdot2013}{2012\cdot2013+1}< 1\)

             \(\frac{2013}{2012}>1\)

\(\Rightarrow\frac{2012\cdot2013}{2012\cdot2013+1}< \frac{2013}{2012}\)

Có : \(\frac{2012}{2013}=1-\frac{2012}{2013}=\frac{2013}{2013}-\frac{2012}{2013}=\frac{1}{2013}\)

         \(\frac{2011}{2012}=1-\frac{2011}{2012}=\frac{2012}{2012}-\frac{2011}{2012}=\frac{1}{2012}\)

Vì \(2013< 2012\)nên \(\frac{1}{2013}< \frac{1}{2012}\)hay \(\frac{2012}{2013}< \frac{2011}{2012}\)

So sánh các phân số sau :

\(\frac{2012\times2013}{2012\times2013+1}\)và \(\frac{2013}{2012}\)

Ta có : 

\(\frac{2012\times2013}{2012\times2013+1}\)

\(=\frac{1}{1}=1\)( cùng rút gọn 2012 x 2013 )

Vì \(1< \frac{2013}{2012}\)nên \(\frac{2012\times2013}{2012\times2013+1}< \frac{2013}{2013}\).

\(\frac{2012}{2013}\)và \(\frac{2011}{2012}\)

Ta có : 

\(1-\frac{2012}{2013}=\frac{1}{2013};1-\frac{2011}{2012}=\frac{1}{2012}\)

Vì \(\frac{1}{2013}< \frac{1}{2012}\)nên \(\frac{2012}{2013}< \frac{2011}{2012}\)

~ Chúc  bạn hok tốt ~

Ta có : 

\(\frac{1}{2013}M=\frac{2013^{2012}+2012}{2013^{2012}+2013}=\frac{2013^{2012}+2013}{2013^{2012}+2013}-\frac{1}{2013^{2012}+2013}=1-\frac{1}{2013^{2012}+2013}\)

Lại có : 

\(\frac{1}{2013}N=\frac{2013^{2011}+2012}{2013^{2011}+2013}=\frac{2013^{2011}+2013}{2013^{2011}+2013}-\frac{1}{2013^{2011}+2013}=1-\frac{1}{2013^{2011}+2013}\)

Vì \(\frac{1}{2013^{2012}+2013}< \frac{1}{2013^{2011}+2013}\) nên \(M=1-\frac{1}{2013^{2012}}>N=1-\frac{1}{2013^{2011}+2013}\)

Vậy \(M>N\)

Chúc bạn học tốt ~ 

Ta có: \(B=\frac{2011}{2012+2013+2014}+\frac{2012}{2012+2013+2014}+\frac{2013}{2012+2013+2014}\)

A= \(\frac{2011}{2012}+\frac{2012}{2013}+\frac{2013}{2014}\)

Xét từng số hạng của A và B

\(\frac{2011}{2012}>\frac{2011}{2012+2013+2014}\)

\(\frac{2012}{2013}>\frac{2012}{2012+2013+2014}\)

\(\frac{2013}{2014}>\frac{2013}{2012+2013+2014}\)

\(\Rightarrow\frac{2011}{2012}+\frac{2012}{2013}+\frac{2013}{2014}>\frac{2011+2012+2013}{2012+2013+2014}\)

\(\Rightarrow A>B\)

Đề bạn ghi có hơi sai chút nên tự tự sửa lại nha!

chờ chút nhé bạn!

khó quá