\(A=1+\dfrac{\dfrac{\left(1+2\right).2}{2}}{2}+\dfrac{\dfrac{\left(1+3\right).3}{2}}{3}+...+\dfrac{\dfrac{\left(1+2013\right).2013}{2}}{2013}\)

\(A=1+\dfrac{\dfrac{3.2}{2}}{2}+\dfrac{\dfrac{4.3}{2}}{3}+...+\dfrac{\dfrac{2014.2013}{2}}{2013}\)

\(A=1+\dfrac{3}{2}+\dfrac{2.3}{3}+...+\dfrac{1007.2013}{2013}\)

\(A=1+\dfrac{3}{2}+2+\dfrac{5}{2}...+1007\)

\(2A=2+3+4+5+6+...+2012+2013+2014\)

\(2A=\dfrac{\left(2+2014\right).2013}{2}\)

\(A=\dfrac{2016.2013}{4}=504.2013\)

\(B=\dfrac{-2}{1.3}+\dfrac{-2}{2.4}+...+\dfrac{-2}{2012.2014}+\dfrac{-2}{2013.2015}\)

\(-B=\dfrac{2}{1.3}+\dfrac{2}{2.4}+...+\dfrac{2}{2012.2014}+\dfrac{2}{2013.2015}\)

\(-B=\left(\dfrac{2}{1.3}+\dfrac{2}{3.5}+...+\dfrac{2}{2013.2015}\right)+\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+...+\dfrac{2}{2012.2014}\right)\)

\(-B=\left(\dfrac{3-1}{1.3}+\dfrac{5-3}{3.5}+...+\dfrac{2015-2013}{2013.2015}\right)+\left(\dfrac{4-2}{2.4}+\dfrac{6-4}{4.6}+...+\dfrac{2014-2012}{2012.2014}\right)\)

\(-B=\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{2013}-\dfrac{1}{2015}\right)+\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}+...+\dfrac{1}{2012}-\dfrac{1}{2014}\right)\)

\(-B=\left(1-\dfrac{1}{2015}\right)+\left(\dfrac{1}{2}-\dfrac{1}{2014}\right)\)

\(-B=\dfrac{2014}{2015}+\dfrac{2012}{2014.2}=\dfrac{2014^2+1006.2015}{2015.2014}\)

\(B=\dfrac{2014^2+1006.2015}{-2015.2014}\)

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 ~

M<4 :)

với cả đây là bài lớp 5

m bé hơn 4

mà giờ lớp 4 học khó nhở

M=2010/2011+2011/2012+2012/2013+2013/2014
4 = 1+1+1+1
Mà 2010/2011 <1
2011/2012<1
2012/2013<1
2013/2014 <1
=>2010/2011+2011/2012+2012/2013+2013/2014​ < 1+1+1+1 
=> M<4

tích cho chị zới nha

Ta thấy 

\(\dfrac{2010}{2011}< 1\)

\(\dfrac{2011}{2012}< 1\)

\(\dfrac{2012}{2013}< 1\)

\(\dfrac{2013}{2014}< 1\) 

=> Tổng M của những phân số trên sẽ nhỏ hơn 1 

=> M < 1

nhận xét

\(\dfrac{2010}{2011}\)<1

...

\(\dfrac{2013}{2014}< 1\)

vì 1<4⇒M<4

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

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

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

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

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

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

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