Đáp án:
\(\frac{{3(n + 1)}}{{n + 3}}\)
Giải thích các bước giải:
\(\begin{array}{l}
(1 + \frac{2}{4})(1 + \frac{2}{{10}})(1 + \frac{2}{{18}})...\frac{{(n + 1)(n + 2)}}{{n(n + 3)}}\\
= \frac{6}{4}.\frac{{12}}{{10}}.\frac{{20}}{{18}}...\frac{{(n + 1)(n + 2)}}{{n(n + 3)}}\\
= \frac{{2.3}}{{1.4}}.\frac{{3.4}}{{2.5}}.\frac{{4.5}}{{3.6}}...\frac{{(n + 1)(n + 2)}}{{n(n + 3)}}\\
= \frac{{2.3.4...(n + 1)}}{{1.2.3...n}}.\frac{{3.4.5...(n + 2)}}{{4.5.6...(n + 3)}}\\
= (n + 1).\frac{3}{{(n + 3)}}\\
= \frac{{3(n + 1)}}{{n + 3}}
\end{array}\)
