Tìm x biết \(\left( {1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{{2013}}} \right)x + 2013 = \frac{{2014}}{1} + \frac{{2015}}{2} + \ldots + \frac{{4025}}{{2012}} + \frac{{4026}}{{2013}}\)
Hãy suy nghĩ và trả lời câu hỏi trước khi xem đáp án
Lời giải:
Báo sai\(\begin{array}{l} \text{Ta có:}\frac{{2014}}{1} + \frac{{2015}}{2} + \ldots + \frac{{4025}}{{2012}} + \frac{{4026}}{{2013}} - 2013 = \left( {\frac{{2014}}{1} - 1} \right) + \left( {\frac{{2015}}{2} - 1} \right) + \ldots + \left( {\frac{{4025}}{{2012}} - 1} \right) + \left( {\frac{{4026}}{{2013}} - 1} \right)\\ = \frac{{2013}}{1} + \frac{{2013}}{2} + \frac{{2013}}{3} + \ldots + \frac{{2013}}{{2012}} + \frac{{2013}}{{2013}} = 2013\left( {\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{{2013}}} \right)\\ Khi\,đó\\ \left( {1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{{2013}}} \right) \cdot x = 2013\left( {\frac{1}{1} + \frac{1}{2} + \ldots + \frac{1}{{2013}}} \right) \Rightarrow x = 2013 \end{array}\)