Tìm x biết \(\left( {\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots .\frac{1}{{2014}}} \right)x = \frac{{2013}}{1} + \frac{{2012}}{2} + \frac{{2011}}{3} \ldots + \frac{1}{{2013}}\)
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Lời giải:
Báo sai\(\begin{array}{l} \left( {\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots .\frac{1}{{2014}}} \right)x = \frac{{2013}}{1} + \frac{{2012}}{2} + \frac{{2011}}{3} \ldots + \frac{1}{{2013}}\\ ta\,có\,\frac{{2013}}{1} + \frac{{2012}}{2} + \frac{{2011}}{3} \ldots + \frac{1}{{2013}} = \left( {1 + \frac{{2012}}{2}} \right) + \left( {1 + \frac{{2011}}{3}} \right) \ldots \left( {1 + \frac{1}{{2013}}} \right) + 1\\ = \frac{{2014}}{2} + \frac{{2014}}{3} + \frac{{2014}}{4} \ldots + \frac{{2014}}{{2013}} + \frac{{2014}}{{2014}} = 2014\left( {\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots .\frac{1}{{2014}}} \right)\\ \Rightarrow \left( {\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots .\frac{1}{{2014}}} \right)x = 2014\left( {\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots .\frac{1}{{2014}}} \right)\\ \Rightarrow x = 2014 \end{array}\)