Ta có `: A = 1/( 1 . 2 . 3 ) + 1/( 2 . 3 . 4 ) + 1/( 3 . 4 . 5 ) + ..... + 1/( a . ( a + 1 ) . ( a + 2 ) )`
`A = ( 2/( 1 . 2 . 3 ) + 2/( 2 . 3 . 4 ) + 2/( 3 . 4 . 5 ) + ..... + 2/( a . ( a + 1 ) . ( a + 2 ) ) ) . 1/2`
Vì `2/( n . ( n + 1 ) . ( n + 2 ) ) = 1/( n . ( n + 1 ) ) - 1/( ( n + 1 ) . ( n + 2 ) )`
`⇒ 2/( 1 . 2 . 3 ) = 1/( 1 . 2 ) - 1/( 2 . 3 ) ; ...... ; 2/( a . ( a + 1 ) . ( a + 2 ) ) = 1/( a . ( a + 1 ) ) - 1/( ( a + 1 ) . ( a + 2 ) )`
` ⇒ A = ( 1/( 1 . 2 ) - 1/( 2 . 3 ) + 1/( 2 . 3 ) - 1/( 3 . 4 ) + 1/( 3 . 4 ) - 1/( 4 . 5 ) + .... + 1/( a . ( a + 1 ) ) - a/( ( a + 1 ) . ( a + 2 ) ) ) . 1/2`
`A = ( 1/( 1 . 2 ) - 1/( ( a + 1 ) . ( a + 2 ) ) ). 1/2`
`A = ( 1/2 - 1/( ( a + 1 ) . ( a + 2 ) ) ) . 1/2`
Vậy `, A = ( 1/2 - 1/( ( a + 1 ) . ( a + 2 ) ) ) . 1/2`