Tìm $x$ , biết: \(\frac{1}{{2.4}} + \frac{1}{{4...

Tìm $x$ , biết: $\frac{1}{{2.4}} + \frac{1}{{4.6}} + \cdot \cdot \cdot + \frac{1}{{(2x - 2).2x}} = \frac{1}{8}\,\,\,\,\,\,\,\,\,\,\,\,\,(x \in \mathbb{N},\,\,x \ge 2)$

$x = -3$ .

$x =3$ .

$x = -2$ .

$x = 2$ .

Đáp án

- Hướng dẫn giải

Phương pháp giải:

Thu gọn vế trái rồi tìm $x$ .

Giải chi tiết:

Ta có:

$\begin{array}{l}\frac{1}{{2.4}} + \frac{1}{{4.6}} + \cdot \cdot \cdot + \frac{1}{{(2x - 2).2x}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{8}\\\frac{1}{2}.\left( {\frac{1}{2} - \frac{1}{4} + \frac{1}{4} - \frac{1}{6} + \cdot \cdot \cdot + \frac{1}{{(2x - 2)}} - \frac{1}{{2x}}} \right) = \frac{1}{8}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{2}.\left( {\frac{1}{2} - \frac{1}{{2x}}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{8}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{2} - \frac{1}{{2x}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{8}:\frac{1}{2}\end{array}$

$\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{2} - \frac{1}{{2x}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{4}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{{2x}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{2} - \frac{1}{4}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{{2x}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{4}\\ \Rightarrow 2x = 4\\\,\,\,\,\,\,\,\,\,x\, = 2\end{array}$