1111-----X------X------X.X------------=?1x2x32X3X44X5X699X100X101前面错误正确的下面：111------+-----+......+-----------1x2x32x3x499x100x101

Question

1 1 1 1
-----X------X------X.X------------=?
1x2x3 2X3X4 4X5X6 99X100X101
前面错误
正确的下面：
1 1 1
------+-----+......+-----------
1x2x3 2x3x4 99x100x101

Answer 1

an=1/(n(n+1)(n+2))
=((1/n)-(1/(n+1))*(1/(n+2))
=(1/(n(n+2))-(1/(n+1)(n+2)))
=(1/2)((1/n)-(1/(n+2))-((1/n+1)-(1/(n+2))
1/(1*2*3)=(1/2)((1/1)-(1/3))-((1/2)-(1/3))
1/(2*3*4)=(1/2)((1/2)-(1/4))-((1/3)-(1/4))
1/(3*4*5)=(1/2)((1/3)-(1/5))-((1/4)-(1/5))
...
1/(99*100*101)=(1/2)((1/99)-(1/101))-((1/100)-(1/101))
所以:
原式=(1/2)(1+(1/2)-(1/100)-(1/101))-((1/2)-(1/101))
=(1/4)+(1/202)-(1/200)
=(1/4)-(1/20200)