Question

1X2X3分之1+2X3X4分之1+3X4X5分之一+。。。。+48X49X50分之一的答案

Answer 1

考察一般项第k项：
1/[k(k+1)(k+2)]=(1/2)[(1/k-1/(k+1) -(1/(k+1)-1/(k+2)]

1/(1×2×3)+1/(2×3×4)+...+1/(48×49×50)
=(1/2)[(1/1-1/2)-(1/2-1/3)+(1/2-1/3)-(1/3-1/4)+...+(1/48-1/49)-(1/49-1/50)]
=(1/2)[(1/1-1/2)-(1/49-1/50)]
=(1/2)(1-1/2-1/49+1/50)
=306/1225

一般的：
1/(1×2×3)+1/(2×3×4)+...+1/[n(n+1)(n+2)]
=(1/2)[(1/1-1/2)-(1/2-1/3)+(1/2-1/3)-(1/3-1/4)+...+(1/n-1/(n+1))-(1/(n+1)-1/(n+2))]
=(1/2)[1-1/2 -1/(n+1)+1/(n+2)]
=(1/2)[1/2 -1/(n+1)+1/(n+2)]
=n(n+3)/[4(n+1)(n+2)]