Đáp án: x=50
Giải thích các bước giải:
$\begin{array}{l}
\frac{3}{{1.5}} + \frac{3}{{5.9}} + ... + \frac{3}{{\left( {4x + 1} \right)\left( {4x + 5} \right)}} = \frac{{153}}{{205}}\left( {dk:x \ne - \frac{5}{4}} \right)\\
\Rightarrow \frac{3}{4}.\left( {\frac{4}{{1.5}} + \frac{4}{{5.9}} + ... + \frac{4}{{\left( {4x + 1} \right)\left( {4x + 5} \right)}}} \right) = \frac{{153}}{{205}}\\
\Rightarrow \frac{{5 - 1}}{{1.5}} + \frac{{9 - 5}}{{5.9}} + ... + \frac{{\left( {4x + 5} \right) - \left( {4x + 1} \right)}}{{\left( {4x + 1} \right)\left( {4x + 5} \right)}} = \frac{{153}}{{205}}:\frac{3}{4}\\
\Rightarrow 1 - \frac{1}{5} + \frac{1}{5} - \frac{1}{9} + ... + \frac{1}{{4x + 1}} - \frac{1}{{4x + 5}} = \frac{{153}}{{205}}.\frac{4}{3}\\
\Rightarrow 1 - \frac{1}{{4x + 5}} = \frac{{204}}{{205}}\\
\Rightarrow \frac{1}{{4x + 5}} = 1 - \frac{{204}}{{205}}\\
\Rightarrow \frac{1}{{4x + 5}} = \frac{1}{{205}}\\
\Rightarrow 4x + 5 = 205\\
\Rightarrow x = 50\left( {tmdk} \right)
\end{array}$
Vậy x=50
