Đáp án:
\[C = \frac{{n + 1}}{{2\left( {3n + 5} \right)}}\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
C = \frac{1}{{2.5}} + \frac{1}{{5.8}} + \frac{1}{{8.11}} + .... + \frac{1}{{\left( {3n + 2} \right)\left( {3n + 5} \right)}}\\
= \frac{1}{3}\left[ {\frac{3}{{2.5}} + \frac{3}{{5.8}} + \frac{3}{{8.11}} + ..... + \frac{3}{{\left( {3n + 2} \right)\left( {3n + 5} \right)}}} \right]\\
= \frac{1}{3}\left[ {\frac{{5 - 2}}{{2.5}} + \frac{{8 - 5}}{{5.8}} + \frac{{11 - 8}}{{8.11}} + .... + \frac{{\left( {3n + 5} \right) - \left( {3n + 2} \right)}}{{\left( {3n + 2} \right)\left( {3n + 5} \right)}}} \right]\\
= \frac{1}{3}.\left[ {\frac{1}{2} - \frac{1}{5} + \frac{1}{5} - \frac{1}{8} + \frac{1}{8} - \frac{1}{{11}} + ..... + \frac{1}{{3n + 2}} - \frac{1}{{3n + 5}}} \right]\\
= \frac{1}{3}.\left[ {\frac{1}{2} - \frac{1}{{3n + 5}}} \right]\\
= \frac{1}{3}.\frac{{3n + 5 - 2}}{{2\left( {3n + 5} \right)}} = \frac{{3n + 3}}{{3.2.\left( {3n + 5} \right)}} = \frac{{n + 1}}{{2\left( {3n + 5} \right)}}
\end{array}\)
