Rút gọn biểu thức: \(\left( {1 - \frac{1}{{{2^2}}}} \right).\left( {1 - \frac{1}{{{3^2}}}} \right).\left( {1 - \frac{1}{{{4^2}}}} \right).....\left( {1 - \frac{1}{{{n^2}}}} \right)\).

Lời giải:

Ta có công thức: \(1 - \frac{1}{{{k^2}}} = \frac{{{k^2} - {1^2}}}{{{k^2}}} = \frac{{\left( {k + 1} \right)\left( {k + 2} \right)}}{{{k^2}}}\)

Áp dụng công thức trên ta được:

\(\left( {1 - \frac{1}{{{2^2}}}} \right)\left( {1 - \frac{1}{{{3^2}}}} \right)\left( {1 - \frac{1}{{{4^2}}}} \right)....\left( {1 - \frac{1}{{{n^2}}}} \right) = \frac{{{2^2} - {1^2}}}{{{2^2}}}.\frac{{{3^2} - {1^2}}}{{{3^2}}}.\frac{{{4^2} - {1^2}}}{{{4^2}}}....\frac{{{n^2} - {1^2}}}{{{n^2}}}\)

\( = \frac{{\left( {2 + 1} \right)\left( {2 - 1} \right)}}{{2.2}}.\frac{{\left( {3 + 1} \right)\left( {3 - 1} \right)}}{{3.3}}.\frac{{\left( {4 + 1} \right)\left( {4 - 1} \right)}}{{4.4}}....\frac{{\left( {n + 1} \right)\left( {n - 1} \right)}}{{n.n}}\)

\( = \frac{{1.3}}{{2.2}} = \frac{{2.4}}{{3.3}} = \frac{{3.5}}{{4.4}}....\frac{{\left( {n + 1} \right)\left( {n - 1} \right)}}{{n.n}} = \frac{{\left[ {1.2.3...\left( {n + 1} \right)} \right].\left[ {3.4.5...\left( {n - 1} \right)} \right]}}{{\left( {2.3.4...n} \right)\left( {2.3.4...n} \right)}}\)

\( = \left( {n + 1} \right).\frac{1}{{2n}} = \frac{{n + 1}}{{2n}}\).