1/2*3+1/3*4+1/4*5+1/5*6*1/6*7+。。。。。+1/99*100
1/3+1/15+1/35+1/63+1/99+1/143

 

一：

1/2*3 = 1/2 - 1/3
1/3*4 = 1/3 - 1/4
...
1/99*100 = 1/99 - 1/100

所以：
1/2*3+1/3*4+1/4*5+1/5*6*1/6*7+。。。。。+1/99*100
＝(1/2 - 1/3 + 1/3 - 1/4 +...+1/99 - 1/100)
＝(1/2 - 1/100)
＝49/100

二：
1/3 = 1/1*3 = (1/1 - 1/3)/2
1/15 = 1/3*5 = (1/3 - 1/5)/2
1/35 = 1/5*7 = (1/5 - 1/7)/2
...
1/143 = 1/11*13 = (1/11 - 1/13)/2

1/3+1/15+1/35+1/63+1/99+1/143 
= (1/1 - 1/3 + 1/3 - 1/5 + ... + 1/11 - 1/13)/2
= (1 - 1/13)/2
= 6/13

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