Đáp án: A<1/25
Giải thích các bước giải:
$\begin{array}{l}
Ta\,có:\frac{n}{{n + 2}} < \frac{{n - 1}}{n}\\
\left( {do:{n^2} < \left( {n + 2} \right)\left( {n - 1} \right) \Rightarrow {n^2} < {n^2} + n - 2\forall n > 2} \right)\\
\Rightarrow A = \frac{1}{3}.\frac{4}{6}.\frac{7}{9}....\frac{{208}}{{210}}\\
\Rightarrow A < \frac{1}{3}.\frac{{4 - 1}}{4}.\frac{{7 - 1}}{7}....\frac{{208 - 1}}{{208}}\\
\Rightarrow A < \frac{1}{3}.\frac{3}{4}.\frac{6}{7}....\frac{{207}}{{208}}\\
\Rightarrow A.A < \frac{{1.4.7...208}}{{3.6.9...210}}.\frac{{1.3.6.9..207}}{{3.4.7.10...208}}\\
\Rightarrow {A^2} < \frac{1}{{210}}.\frac{1}{3}\\
\Rightarrow {A^2} < \frac{1}{{630}} < \frac{1}{{625}} = {\left( {\frac{1}{{25}}} \right)^2}\\
\Rightarrow A < \frac{1}{{25}}
\end{array}$
