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1/n(n+1)(n+2)
=1/n(n+1)-1/n(n+2)
=1/n-1/(n+1)- 1/2[1/n-1/(n+2)]
=(1/2)[1/n+1/(n+2)]-1/(n+1)
1/6*7*8=(1/2)(1/6+1/8)-1/7
1/7*8*9=(1/2)(1/7+1/9)-1/8
1/8*9*10=(1/2)(1/8+1/10)-1/9
---
1/48*49*50=(1/2)(1/48+1/50)-1/49
所以：1/6*7*8+1/7*8*9+1/8*9*10+...+1/48*49*50
=(1/2)(1/6+1/8+1/7+1/9+1/8+1/10+---+1/48+1/50)-(1/7+1/8+1/9+--+1/49)
=(1/2)(1/6-1/7)+(1/2)(1/50-1/49)=(1/2)((1/6-1/7+1/50-1/49)
=(1/2)*(1/42-1/2450)
=1/n(n+1)-1/n(n+2)
=1/n-1/(n+1)- 1/2[1/n-1/(n+2)]
=(1/2)[1/n+1/(n+2)]-1/(n+1)
1/6*7*8=(1/2)(1/6+1/8)-1/7
1/7*8*9=(1/2)(1/7+1/9)-1/8
1/8*9*10=(1/2)(1/8+1/10)-1/9
---
1/48*49*50=(1/2)(1/48+1/50)-1/49
所以：1/6*7*8+1/7*8*9+1/8*9*10+...+1/48*49*50
=(1/2)(1/6+1/8+1/7+1/9+1/8+1/10+---+1/48+1/50)-(1/7+1/8+1/9+--+1/49)
=(1/2)(1/6-1/7)+(1/2)(1/50-1/49)=(1/2)((1/6-1/7+1/50-1/49)
=(1/2)*(1/42-1/2450)
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