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\(B=\frac{2013-1}{2013}+\frac{2014-1}{2014}+\frac{2012+3}{2012}\)
\(B=1-\frac{1}{2013}+1-\frac{1}{2014}+1+\frac{3}{2012}=3+\frac{3}{2012}-\left(\frac{1}{2013}+\frac{1}{2014}\right)\)
Ta có
\(\frac{1}{2013}< \frac{1}{2012};\frac{1}{2014}< \frac{1}{2012}\Rightarrow\frac{1}{2013}+\frac{1}{2014}< \frac{2}{2012}\)
Mà \(\frac{3}{2012}-\frac{2}{2012}=\frac{1}{2012}>0\Rightarrow\frac{3}{2012}-\left(\frac{1}{2013}+\frac{1}{2014}\right)>0\)
=> B>3
$\frac{\frac{2010}{2011}}{\frac{2012}{2013}}+\frac{\frac{2011}{2012}}{\frac{2013}{2014}}+\frac{\frac{2012}{2013}}{\frac{2014}{2015}}$
$\frac{\frac{2010}{2011}}{\frac{2012}{2013}}+\frac{\frac{2011}{2012}}{\frac{2013}{2014}}+\frac{\frac{2012}{2013}}{\frac{2014}{2015}}$
$\frac{\frac{2010+2011+2012}{2011+2012+2013}}{\frac{2012+2013+2014}{2013+2014+2015}}$
$\frac{\frac{2010}{2011}+\frac{2011}{2012}+\frac{2012}{2013}}{\frac{2012+2013+2014}{2013+2014+2015}}$
$\frac{\frac{2010+2011+2012}{2011+2012+2013}}{\frac{2012}{2013}+\frac{2013}{2014}+\frac{2014}{2015}}$
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 B = 2013 . 2013
=> B = 2013.(2014 - 1)
=> B = 2013 . 2014 - 2013
=> B = (2012+1).2014 - 2013
=> B = 2012.2014 + 2014 - 2013
=> B = 2012.2014 + 1 > 2012.2014 = A
Vậy A<B
\(B=\frac{2012}{2013}+\frac{2013}{2014}+\frac{2015}{2012}\)
\(B=\frac{2012}{2013}+\frac{2013}{2014}+\left(\frac{1}{2012}+\frac{1}{2012}+\frac{2013}{2012}\right)\)
\(B=\left(\frac{2012}{2013}+\frac{1}{2012}\right)+\left(\frac{2013}{2014}+\frac{1}{2012}\right)+\frac{2013}{2012}\)
\(3=1+1+1\)
\(\frac{2012}{2013}+\frac{1}{2012}>1\)
\(\frac{2013}{2014}+\frac{1}{2012}>1\)
\(\frac{2013}{2012}>1\)
vậy B > 3