Lời giải

Đáp án đúng là: A

${\rm{A}} = \left( {1 - \frac{1}{{{2^2}}}} \right)\left( {1 - \frac{1}{{{3^2}}}} \right)...\left( {1 - \frac{1}{{{n^2}}}} \right)$

$= \frac{{{2^2} - 1}}{{{2^2}}} \cdot \frac{{{3^2} - 1}}{{{3^2}}} \cdot \frac{{{4^2} - 1}}{{{4^2}}} \cdot \frac{{{5^2} - 1}}{{{5^2}}} \cdots \frac{{{n^2} - 1}}{{{n^2}}}$

$= \frac{{1.3}}{{{2^2}}}.\frac{{2.4}}{{{3^2}}}.\frac{{3.5}}{{{4^2}}}.\frac{{4.6}}{{{5^2}}}...\frac{{\left( {n - 1} \right)\left( {n + 1} \right)}}{{{n^2}}}$

$= \frac{{1.2.3.4...\left( {n - 1} \right)}}{{2.3.4.5...n}}.\frac{{3.4.5.6...\left( {n + 1} \right)}}{{2.3.4.5...n}}$

$= \frac{1}{n}.\frac{{n + 1}}{2} = \frac{{n + 1}}{{2n}}$