\(\begin{array}{l}
a)\,\,\left( {\dfrac{1}{2} - 1} \right)\left( {\dfrac{1}{3} - 1} \right)...\left( {\dfrac{1}{{2002}} - 1} \right)\left( {\dfrac{1}{{2003}} - 1} \right)\\
= \dfrac{{ - 1}}{2}.\dfrac{{ - 2}}{3}...\dfrac{{ - 2001}}{{2002}}.\dfrac{{ - 2002}}{{2003}}\\
= \dfrac{{\left( { - 1} \right).\left( { - 2} \right)...\left( { - 2001} \right).\left( { - 2002} \right)}}{{2.3...2002.2003}}\\
= \dfrac{{ - 1}}{{2003}}.\dfrac{{\left( { - 2} \right).\left( { - 3} \right)...\left( { - 2001} \right).\left( { - 2002} \right)}}{{2.3...2001.2002}}\\
= \dfrac{{ - 1}}{{2003}}.\left( { - 1} \right) = \dfrac{1}{{2003}}
\end{array}\)
(Vì dãy \(2 ; 3 ; ...; 2001; 2002\) có \((2002 - 2) : 1 + 1 = 2001\) số hạng nên tích \({\left( { - 2} \right).\left( { - 3} \right)...\left( { - 2001} \right).\left( { - 2002} \right)} = -{2.3...2001.2002}\))
\(\begin{array}{l}
b)\,\,\left( { - 1\dfrac{1}{2}} \right)\left( { - 1\dfrac{1}{3}} \right)...\left( { - 1\dfrac{1}{{2003}}} \right)\left( { - 1\dfrac{1}{{2004}}} \right)\\
= \dfrac{{ - 3}}{2}.\dfrac{{ - 4}}{3}...\dfrac{{ - 2004}}{{2003}}.\dfrac{{ - 2005}}{{2004}}\\
= \dfrac{{\left( { - 3} \right).\left( { - 4} \right)...\left( { - 2004} \right).\left( { - 2005} \right)}}{{2.3....2003.2004}}\\
= \dfrac{{ - 2005}}{2}.\dfrac{{\left( { - 3} \right).\left( { - 4} \right)...\left( { - 2003} \right).\left( { - 2004} \right)}}{{3.4....2003.2004}}\\
= \dfrac{{ - 2005}}{2}.1 = \dfrac{{ - 2005}}{2}
\end{array}\)
(Vì dãy \(3 ; 4 ; ...; 2003; 2004\) có \((2004 - 3) : 1 + 1 = 2002\) số hạng nên tích \({\left( { - 3} \right).\left( { - 4} \right)...\left( { - 2003} \right).\left( { - 2004} \right)} = {2.3...2001.2002}\))
