A=1/2 + 5/6 + 11/12 + 19/20 + 29/30 + 41/42 + 55/56 + 71/72 + 89/90
= 1 - 1/2 + 1 - 1/6 + 1-1/12 + 1 - 1/20 + 1 - 1/30 + 1 - 1/42 + 1 - 1/56 + 1/72 + 1 - 1/90
= ( 1 + 1 + 1 + 1 + 1 +1 + 1 + 1 +1 ) - ( 1/2 + 1/6 + 1/12 + 1/20 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90 )
= 9 - ( 1 / 1.2 + 1 / 2.3 + ... 1/9.10 )
= 9 - ( 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/9 - 1/10 )
= 9 - ( 1 - 1/10 )
= 9 - 9/10
= 81/10
