Tính \(\frac{{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots + \frac{1}{{99}} + \frac{1}{{100}}}}{{\frac{{99}}{1} + \frac{{98}}{2} + \frac{{97}}{3} + \ldots + \frac{1}{{99}}}}\)
Hãy suy nghĩ và trả lời câu hỏi trước khi xem đáp án
Lời giải:
Báo sai\(\begin{array}{l} Ta\,có\,:\frac{{99}}{1} + \frac{{98}}{2} + \frac{{97}}{3} + \ldots + \frac{1}{{99}}\\ = 99 + \frac{{98}}{2} + \frac{{97}}{3} + \ldots + \frac{1}{{99}}\\ = \left( {1 + \frac{{98}}{2}} \right) + \left( {1 + \frac{{97}}{3}} \right) + \ldots + \left( {1 + \frac{1}{{99}}} \right) + 1 = \frac{{100}}{2} + \frac{{100}}{3} + \ldots + \frac{{100}}{{99}} + \frac{{100}}{{100}}\\ = 100\left( {\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots + \frac{1}{{100}}} \right)\\ \Rightarrow A = \frac{{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots + \frac{1}{{100}}}}{{\frac{{99}}{1} + \frac{{98}}{2} + \frac{{97}}{3} + \ldots + \frac{1}{{99}}}} = \frac{{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots + \frac{1}{{100}}}}{{100\left( {\frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{{100}}} \right)}} = \frac{1}{{100}} \end{array}\)
