Đáp án:

$D=\dfrac{9}{20}\\ E=-\dfrac{97}{99}\\  E=\dfrac{10}{21}\\ F-\dfrac{16}{33}.$

Giải thích các bước giải:

$D=\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+\dots+\dfrac{1}{19}{20}\\ =\dfrac{3-2}{2.3}+\dfrac{4-3}{3.4}+\dfrac{5-4}{4.5}+\dots+\dfrac{20-19}{19}{20}\\ =\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dots+\dfrac{1}{19}-\dfrac{1}{20}\\ =\dfrac{1}{2}-\dfrac{1}{20}\\ =\dfrac{9}{20}\\ E=\dfrac{1}{99}-\dfrac{1}{99.98}-\dfrac{1}{98.97}-\dots-\dfrac{1}{2.1}\\ =\dfrac{1}{99}-\left(\dfrac{1}{99.98}+\dfrac{1}{98.97}+\dots+\dfrac{1}{2.1}\right)\\ =\dfrac{1}{99}-\left(\dfrac{1}{1.2}+\dots+\dfrac{1}{97.98}+\dfrac{1}{98.99}\right)\\ =\dfrac{1}{99}-\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dots+\dfrac{1}{97}-\dfrac{1}{98}+\dfrac{1}{98}-\dfrac{1}{99}\right)\\ =\dfrac{1}{99}-\left(1-\dfrac{1}{99}\right)\\ =-\dfrac{97}{99}\\ E=\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+\dots+\dfrac{1}{19.21}\\ 2E=\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+\dots+\dfrac{2}{19.21}\\ =\dfrac{3-1}{1.3}+\dfrac{5-3}{3.5}+\dfrac{7-5}{5.7}+\dots+\dfrac{21-19}{19.21}\\ =1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dots+\dfrac{1}{19}-\dfrac{1}{21}\\ =1-\dfrac{1}{21}\\ =\dfrac{20}{21}\\ \Rightarrow E=\dfrac{10}{21}\\ F=\dfrac{1}{99}-\dfrac{1}{99.97}-\dfrac{1}{97.95}-\dots-\dfrac{1}{3.1}\\ =\dfrac{1}{99}-\left(\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dots+\dfrac{1}{97.99}\right)\\ =\dfrac{1}{99}-\dfrac{1}{2}\left(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dots+\dfrac{2}{97.99}\right)\\ =\dfrac{1}{99}-\dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dots+\dfrac{1}{97}-\dfrac{1}{99}\right)\\ =\dfrac{1}{99}-\dfrac{1}{2}\left(1-\dfrac{1}{99}\right)\\ =\dfrac{1}{99}-\dfrac{49}{99}\\ =-\dfrac{16}{33}.$