`1/(1.2) + 1/(2.3) + ... + 1/(1999.2000)`
`= 1 - 1/2 + 1/2 - 1/3 + ... + 1//1999 - 1/2000`
`= 1 - 1/2000`
`= 1999/2000`
`1/(1.4) + 1/(4.7) + ... + 1/(100 . 103)`
`= 1/3 . (3/(1.4) + 3/(4.7) + ... + 3/(100 . 103))`
`= 1/3 . (1 - 1/4 + 1/4 - 1/7 + ... + 1/100 - 1/103)`
`= 1/3( 1 - 1/103)`
`= 1/3 . 102/103`
`= 34/103`
`8/9 - 1/72 - 1/56 - ... - 1/6 - 1/2`
`= 8/9 - (1/2 + 1/6 + ... + 1/72)`
`= 8/9 - (1/(1.2) + 1/(2.3) + ... + 1/(8.9))`
`= 8/9 - (1- 1/2 + 1/2 - 1/3 + ... + 1/8 - 1/9)`
`= 8/9 - (1- 1/9)`
`= 8/9 - 8/9 `
`= 0`
c) \(\dfrac{8}{9}-\dfrac{1}{72}-\dfrac{1}{56}-\dfrac{1}{42}-\dfrac{1}{6}-\dfrac{1}{2}\)
= \(-\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{6.7}+\dfrac{1}{7.8}+\dfrac{1}{8.9}\right)+\dfrac{8}{9}\)
= \(-\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{6}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{8}+\dfrac{1}{8}-\dfrac{1}{9}\right)+\dfrac{8}{9}\)
= \(-\left(1-\dfrac{1}{9}\right)+\dfrac{8}{9}\)
= 0
b) Có: \(\dfrac{1}{1.4}+\dfrac{1}{4.7}+\dfrac{1}{7.10}+...+\dfrac{1}{100.103}\)
= \(3\left(\dfrac{1}{1.4}+\dfrac{1}{4.7}+\dfrac{1}{7.10}+...+\dfrac{1}{100.103}\right):3\)
= \(\left(\dfrac{3}{1.4}+\dfrac{3}{4.7}+\dfrac{3}{7.10}+...+\dfrac{3}{100.103}\right):3\)
= \(\left(1-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+\dfrac{1}{10}-...-\dfrac{1}{100}+\dfrac{1}{100}-\dfrac{1}{103}\right):3\)
=\(\left(1-\dfrac{1}{103}\right):3\)
=\(\dfrac{102}{103}:3=\dfrac{34}{103}\)