$A = 1/2 + 5/6 + 11/12 + 19/20 + 29/30 + 41/42 + 55/56 + 71/72 + 89/90$
$⇔A = (1 - 1/2)+ (1- 1/6) + (1 - 1/12) +(1-1/20) + (1 - 1/30) + (1 - 1/42) + (1 - 1/56) + (1-1/72) + (1 -1/90)$
$⇔A=9 - (1/2 + 1/6 + 1/12 + 1/20 + 1/30 +1/42 + 1/56 + 1/72 + 1/90)$
$⇔A = 9- (1/1.2 + 1/2.3 + 1/3.4 + 1/4.5 + 1/5.6 + 1/6.7+ 1/7.8 + 1/8.9 + 1/9.10)$
$⇔ A =9 - (1 - 1/2 +1/2 - 1/3+ 1/3 - 1/4 + 1/4 - 1/5 + 1/5 - 1/6 + 1/6 - 1/7 + 1/7 - 1/8 + 1/8 - 1/9 + 1/9 -1/10)$
$⇔A = 9 - (1 - 1/10)$
$⇔A = 9 - 9/10$
$⇔A = 90/10 - 9/10$
$⇔A = 81/10$.
