Question

1/1+（2/1-1/2）+（3/1-2/2+1/3）+（4/1-3/2+2/3-1/4）+。。。+（9/1-8/2+7/3-6/4+。。。+3/7-2/8+1/9）

Answer 1

原式
= (1+2+3+...+9)/1 - (1+2+3+...+8)/2 + (1+2+3+...+7)/3 - (1+2+3+...+6)/4 +……+ 1/9
= 45/1 - 36/2 + 28/3 - 21/4 + 15/5 - 10/6 + 6/7 - 3/8 + 1/9
= 45 - 18 + 28/3 - 21/4 + 3 - 5/3 + 6/7 - 3/8 + 1/9
= (45-18+3) + (28/3-5/3) -(21/4+3/8) + 6/7 + 1/9
= 30 + (7+2/3) -(5+5/8) + (1-1/7) + 1/9
= (30+7-5+1) + (2/3-5/8-1/7+1/9)
= 33 + (336-315-72+56)/504
= 33又5/504

Answer 2

原式
= (1+2+3+...+9)/1 - (1+2+3+...+8)/2 + (1+2+3+...+7)/3 - (1+2+3+...+6)/4 +……+ 1/9
= 45/1 - 36/2 + 28/3 - 21/4 + 15/5 - 10/6 + 6/7 - 3/8 + 1/9
= 45 - 18 + 28/3 - 21/4 + 3 - 5/3 + 6/7 - 3/8 + 1/9
= (45-18+3) + (28/3-5/3) -(21/4+3/8) + 6/7 + 1/9
= 30 + (7+2/3) -(5+5/8) + (1-1/7) + 1/9
= (30+7-5+1) + (2/3-5/8-1/7+1/9)
= 33 + (336-315-72+56)/504
= 33又5/504