Question

2/1*2*3+2/2*3*4+2/3*4*5+...+2/2011*2012*2013=?

Answer 1

请问题目是这个吗？
2/(1*2*3)+...+2/(2011*2012*2013)
还是2/1*2*3+...+2/2011*2012*2013？
两者有区别的，2/1*2*3=2*2*3=12，2/(1*2*3)=2/6=1/3，到底是哪一个？

2/[（n-1）n（n+1）]=[(n+1)-(n-1)]/[（n-1）n（n+1）]=(n+1)/[（n-1）n（n+1）]-(n-1)/[（n-1）n（n+1）]=1/[n(n-1)]-1/[n(n+1)]
2/(1*2*3)+...+2/(2011*2012*2013)==1/(1*2)-1/(2*3)+1/(2*3)-1/(3*4)+1/(3*4)-1/(4*5)+...+1/(2011*2012)-1/(2012*2013)=1/1*2-1/2012*2013=1/2-1/4050156=2025077/4050156

追问

是2/(1*2*3)+...+2/(2011*2012*2013)

追答

2/[（n-1）n（n+1）]=[(n+1)-(n-1)]/[（n-1）n（n+1）]=(n+1)/[（n-1）n（n+1）]-(n-1)/[（n-1）n（n+1）]=1/[n(n-1)]-1/[n(n+1)]
2/(1*2*3)+...+2/(2011*2012*2013)==1/(1*2)-1/(2*3)+1/(2*3)-1/(3*4)+1/(3*4)-1/(4*5)+...+1/(2011*2012)-1/(2012*2013)=1/1*2-1/2012*2013=1/2-1/4050156=2025077/4050156

Answer 2

2/(1*2*3)+2/(2*3*4)+2/(3*4*5)......+2/(2011*2012*2013)
=(1/1*2-1/2*3)+(1/2*3-1/3*4)+(1/3*4-1/4*5)+...+(1/2011*2012-1/2012*2013)
=1/1*2-1/2*3+1/2*3-1/3*4+1/3*4-1/4*5+...+1/2011*2012-1/2012*2013
=1/1*2-1/2012*2013
=1/2-1/4050156
=2025077/4050156