Đáp án:

 Bài 9. 
$\frac{{103}}{{303}}$

Giải thích các bước giải:

Bài 9

$\begin{array}{l}
B = \left( {1 - \frac{2}{{2 \times 3}}} \right) \times \left( {1 - \frac{2}{{3 \times 4}}} \right) \times \left( {1 - \frac{2}{{4 \times 5}}} \right) \times ... \times \left( {1 - \frac{2}{{101 \times 102}}} \right)\\
B = \frac{4}{6} \times \frac{{10}}{{12}} \times \frac{{18}}{{20}} \times ... \times \frac{{10300}}{{10302}}\\
B = \frac{{1 \times 4}}{{2 \times 3}} \times \frac{{2 \times 5}}{{3 \times 4}} \times \frac{{3 \times 6}}{{4 \times 5}} \times ... \times \frac{{100 \times 103}}{{101 \times 102}}\\
B = \frac{{1 \times 2 \times 3 \times ... \times 100}}{{2 \times 3 \times 4 \times ... \times 101}} \times \frac{{4 \times 5 \times 6 \times ... \times 103}}{{3 \times 4 \times 5 \times ... \times 102}}\\
B = \frac{1}{{101}} \times \frac{{103}}{3} = \frac{{103}}{{303}}
\end{array}$

\[\begin{array}{l}
B = \left( {1 - \frac{1}{{2.3}}} \right) \times \left( {1 - \frac{2}{{3.4}}} \right) \times \left( {1 - \frac{2}{{4.5}}} \right) \times ... \times \left( {1 - \frac{2}{{101.102}}} \right)\\
B = \frac{4}{6} \times \frac{{10}}{{12}} \times \frac{{18}}{{20}} \times ... \times \frac{{10300}}{{10302}}\\
B = \frac{{1.4}}{{2.3}} \times \frac{{2.5}}{{3.4}} \times \frac{{3.6}}{{4.5}} \times ... \times \frac{{100 \times 103}}{{101.102}}\\
B = \frac{{1.2.3.\,\,...\,\,.100}}{{2.3.4.\,\,...\,\,.101}} \times \frac{{4.5.6.\,\,...\,\,.103}}{{3.4.5.\,\,...\,\,.102}}\\
B = \frac{1}{{101}} \times \frac{{103}}{3} = \frac{{103}}{{303}}
\end{array}\]