Tìm x biết \(\frac{1}{{14}} + \frac{1}{{35}} + \frac{1}{{65}} + \ldots + \frac{2}{{{x^2} + 3x}} = \frac{1}{9}\)
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Lời giải:
Báo sai\(\begin{array}{l} \frac{1}{{14}} + \frac{1}{{35}} + \frac{1}{{65}} + \ldots + \frac{2}{{{x^2} + 3x}} = \frac{1}{9}\\ \Rightarrow \frac{2}{{28}} + \frac{2}{{70}} + \frac{2}{{130}} + \ldots + \frac{2}{{x(x + 3)}} = \frac{1}{9}\\ \Rightarrow \frac{2}{{4.7}} + \frac{2}{{7.10}} + \frac{2}{{10.13}} + \ldots + \frac{2}{{x(x + 3)}} = \frac{1}{9}\\ \Rightarrow \frac{2}{3}\left( {\frac{3}{{4.7}} + \frac{3}{{7.10}} + \ldots + \frac{3}{{x(x + 3)}}} \right) = \frac{1}{9} = \frac{2}{3}\left( {\frac{1}{4} - \frac{1}{{x + 3}}} \right) = \frac{1}{9}\\ \Rightarrow \frac{1}{4} - \frac{1}{{x + 3}} = \frac{1}{6}\\ \Rightarrow \frac{1}{4} - \frac{1}{6} = \frac{1}{{x + 3}}\\ \Rightarrow \frac{1}{{12}} = \frac{1}{{x + 3}}\\ \Rightarrow x = 9 \end{array}\)
