(2012 * 2013 + 2013 * 2014 ) * ( 1 + một phần hai : 1 một phần hai - 1 một phần ba )
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\(A=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}}{\dfrac{2013}{1}+\dfrac{2012}{2}+...+\dfrac{1}{2013}}\)
\(=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}}{\left(\dfrac{2012}{2}+1\right)+\left(\dfrac{2011}{3}+1\right)+...+\left(\dfrac{1}{2013}+1\right)+\dfrac{2014}{2014}}\)
\(=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}}{2014\left(\dfrac{1}{2}+\dfrac{1}{.3}+...+\dfrac{1}{2014}\right)}\)
\(=\dfrac{1}{2014}\)
\(\frac{2013}{2012}-\frac{2014}{2013}-\frac{1}{2012.2013}\)
= \(\left(\frac{2012}{2012}+\frac{1}{2012}\right)-\left(\frac{2013}{2013}+\frac{1}{2013}\right)-\frac{1}{2012.2013}\)
= \(1+\frac{1}{2012}-1-\frac{1}{2013}-\frac{1}{2012+2013}\)
= \(\frac{1}{2012}-\frac{1}{2013}-\frac{1}{2012.2013}\)
= \(\frac{1}{2012.2013}-\frac{1}{2012.2013}\)
= 0
\(\frac{2013}{2012}-\frac{2014}{2013}-\frac{1}{2012.2013}\)
\(A=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}}{\frac{2013}{1}+\frac{2012}{2}+\frac{2011}{3}+...+\frac{1}{2013}}\)
\(A=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}}{\left(\frac{2012}{2}+1\right)+\left(\frac{2011}{3}+1\right)+...+\left(\frac{1}{2013}+1\right)+1}\)
\(A=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}}{\frac{2014}{2}+\frac{2014}{3}+...+\frac{2014}{2013}+\frac{2014}{2014}}\)
\(A=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}}{2014.\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2014}\right)}\)\
\(A=\frac{1}{2014}\)
A=\(\dfrac{28}{15}\)(\(\dfrac{1}{2}\))\(^2\).3+(\(\dfrac{8}{15}-\dfrac{79}{60}\)):\(\dfrac{47}{24}\) A=28\(\dfrac{28}{15}.\dfrac{1}{4}.3+\dfrac{\left(-47\right)}{60}.\dfrac{24}{47}\) A=\(\dfrac{7}{5}+\dfrac{\left(-2\right)}{5}=\dfrac{5}{5}=1\) Vậy A=1
Tổng S có 50 phân số
=> S > 1/100 + 1/100 + 1/100 +...+ 1/100 (50 phân số) => S > 1/2.
Vậy S > 1/2