a.
\(S=\dfrac{1}{2022}-\dfrac{5}{2}\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+...+\dfrac{2}{2020.2022}\right)\)
\(=\dfrac{1}{2022}-\dfrac{5}{2}\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{2020}-\dfrac{1}{2022}\right)\)
\(=\dfrac{1}{2022}-\dfrac{5}{2}\left(\dfrac{1}{2}-\dfrac{1}{2022}\right)\)
\(=-\dfrac{1262}{1011}\)
b.
\(A=\dfrac{4.5}{5.7.31}+\dfrac{6.5}{7.5.41}+\dfrac{5.9}{5.10.41}+\dfrac{7.5}{5.10.57}\)
\(=5\left(\dfrac{4}{31.35}+\dfrac{6}{35.41}+\dfrac{9}{41.50}+\dfrac{7}{50.57}\right)\)
\(=5\left(\dfrac{1}{31}-\dfrac{1}{35}+\dfrac{1}{35}-\dfrac{1}{41}+\dfrac{1}{41}-\dfrac{1}{50}+\dfrac{1}{50}-\dfrac{1}{57}\right)\)
\(=5\left(\dfrac{1}{31}-\dfrac{1}{57}\right)\)
Tương tự:
\(B=2\left(\dfrac{7}{31.38}+\dfrac{5}{38.43}+\dfrac{3}{43.46}+\dfrac{11}{46.57}\right)\)
\(=2\left(\dfrac{1}{31}-\dfrac{1}{38}+\dfrac{1}{38}-\dfrac{1}{43}+\dfrac{1}{43}-\dfrac{1}{46}+\dfrac{1}{46}-\dfrac{1}{57}\right)\)
\(=2\left(\dfrac{1}{31}-\dfrac{1}{57}\right)\)
\(\Rightarrow\dfrac{A}{B}=\dfrac{5}{2}\)
