`a) 1/( 1 . 2 ) + 1/( 2 . 3 ) + 1/( 3 . 4 ) + ..... + 1/( 1999 . 2000 )`
`= 1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + .... + 1/1999 - 1/2000`
`= 1 - 1/2000`
`= 1999/2000`
`b) 1/( 1 . 4 ) + 1/( 4 . 7 ) + 1/( 7 . 10 ) + ..... + 1/( 100 . 103 )`
`= 1/3 . ( 3/( 1 . 4 ) + 3/( 4 . 7 ) + 3/( 7 . 10 ) + ..... + 3/( 100 . 103 ) )`
`= 1/3 . ( 1/1 - 1/4 + 1/4 - 1/7 + 1/7 - 1/10 + ..... + 1/100 - 1/103 )`
`= 1/3 . ( 1 - 1/103 )`
`= 1/3 . 102/103`
`= 34/103`
`c) 8/9 - 1/72 - 1/56 - 1/42 - ..... - 1/6 - 1/2`
`= 8/9 - ( 1/72 + 1/56 + 1/42 + ..... + 1/6 + 1/2 )`
`= 8/9 - ( 1/2 + 1/6 + ..... + 1/56 + 1/72 )`
`= 8/9 - ( 1/( 1 . 2 ) + 1/( 2 . 3 ) + .... + 1/( 7 . 8 ) + 1/( 8 . 9 ) )`
`= 8/9 - ( 1/1 - 1/2 + 1/2 - 1/3 + .... + 1/7 - 1/8 + 1/8 - 1/9 )`
`= 8/9 - ( 1 - 1/9 )`
`= 8/9 - 8/9`
`= 0`
