a: \(A=\dfrac{2}{1\cdot3}+\dfrac{2}{3\cdot5}+...+\dfrac{2}{2021\cdot2023}\)
\(=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{2021}-\dfrac{1}{2023}\)
\(=1-\dfrac{1}{2023}=\dfrac{2022}{2023}\)
b: \(B=\dfrac{1}{2\cdot5}+\dfrac{1}{5\cdot8}+...+\dfrac{1}{95\cdot98}\)
\(=\dfrac{1}{3}\left(\dfrac{3}{2\cdot5}+\dfrac{3}{5\cdot8}+...+\dfrac{3}{95\cdot98}\right)\)
\(=\dfrac{1}{3}\left(\dfrac{1}{2}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{8}+...+\dfrac{1}{95}-\dfrac{1}{98}\right)\)
\(=\dfrac{1}{3}\left(\dfrac{1}{2}-\dfrac{1}{98}\right)=\dfrac{1}{3}\cdot\dfrac{48}{98}=\dfrac{16}{98}=\dfrac{8}{49}\)
c: \(C=\dfrac{1}{8}+\dfrac{1}{24}+\dfrac{1}{48}+...+\dfrac{1}{9800}\)
\(=\dfrac{1}{2\cdot4}+\dfrac{1}{4\cdot6}+\dfrac{1}{6\cdot8}+...+\dfrac{1}{98\cdot100}\)
\(=\dfrac{1}{2}\left(\dfrac{2}{2\cdot4}+\dfrac{2}{4\cdot6}+...+\dfrac{2}{98\cdot100}\right)\)
\(=\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{98}-\dfrac{1}{100}\right)\)
\(=\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{100}\right)=\dfrac{1}{2}\cdot\dfrac{49}{100}=\dfrac{49}{200}\)
d: \(D=\dfrac{2}{15}+\dfrac{2}{35}+...+\dfrac{2}{399}\)
\(=\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+...+\dfrac{2}{19\cdot21}\)
\(=\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{19}-\dfrac{1}{21}\)
\(=\dfrac{1}{3}-\dfrac{1}{21}=\dfrac{6}{21}=\dfrac{2}{7}\)
`A=2/(1.3) + 2/(3.5) + ... + 2/(2021 . 2023)`
`=1 - 1/3 + 1/3 - 1/5 + ... + 1/2021 - 1/2023`
`= 1 - 1/2023`
`= 2022/2023`
`B = 1/(2.5) + 1/(5.8) + 1/(8.11) + ... + 1/(92 . 95) + 1/(95.98)`
`=> 3B = 3/(2.5) + 3/(5.8) + ... + 3/(95.98)`
`=> 3B = 1/2 - 1/5 + 1/5 - 1/8 + ... + 1/95 - 1/98`
`=> 3B = 1/2 - 1/98`
`=> 3B = 24/49`
`=> B = 8/49`
`C = 1/8 + 1/24 + ... + 1/9800`
`=> C = 1/(2.4) + 1/(4.6) + 1/(6.8) + .. + 1/(98.100)`
`=> 2C = 1/2 - 1/4 +... + 1/98 - 1/100`
`=> 2C = 1/2 - 1/100`
`=> 2C = 49/100`
`=> C = 49/200`
`D = 2/15 + 2/35 + 2/63 + .. + 2/399`
`=> D = 2/(3.5) + 2/(5.7) + 2/(7.9) + ... + 2/(19.21)`
`=> D = 1/3 - 1/5 + ... + 1/19 - 1/21`
`=> D = 1/3 - 1/21 = 2/7`
