a,\(\left(\dfrac{1}{2}+1\right)\left(\dfrac{1}{3}+1\right)\left(\dfrac{1}{4}+1\right)...\left(\dfrac{1}{999}+1\right)\)
\(=\dfrac{3}{2}.\dfrac{4}{3}.\dfrac{5}{4}.....\dfrac{1000}{999}\)
\(=\dfrac{3.4.5....1000}{2.3.4....999}=\dfrac{1000}{2}=500\)
b,\(\left(\dfrac{1}{2}-1\right)\left(\dfrac{1}{3}-1\right)\left(\dfrac{1}{4}-1\right)...\left(\dfrac{1}{1000}-1\right)\)
\(=\dfrac{-1}{2}.\dfrac{-2}{3}.\dfrac{-3}{4}.....\dfrac{-999}{1000}\)
=\(\dfrac{-\left(1.2.3....999\right)}{2.3.4....1000}=\dfrac{-1}{1000}\)
c,\(\dfrac{3}{2^2}.\dfrac{8}{3^2}.\dfrac{15}{4^2}....\dfrac{99}{10^2}\)
\(=\dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}.\dfrac{3.5}{4.4}....\dfrac{9.11}{10.10}\)
\(=\dfrac{1.3.2.4.3.5....9.11}{2.2.3.3.4.4....10.10}\)
\(=\dfrac{1.2.3...9}{2.3.4...10}.\dfrac{3.4.5...11}{2.3.4...10}\)
\(=\dfrac{1}{10}.\dfrac{11}{2}=\dfrac{11}{20}\)
