试证:对任意的正整数n,有1/1*2*3+1/2*3*4+……+1/n(n+1)*(n+2)

问题描述:

试证:对任意的正整数n,有1/1*2*3+1/2*3*4+……+1/n(n+1)*(n+2)

1/1*2*3+1/2*3*4+……+1/n*(n+1)*(n+2)
=1/2*[1/1*2-1/2*3+1/2*3-1/3*4+……+1/n(n+1)-1/(n+1)*(n+2)]
=1/2*[1/1*2-1/(n+1)(n+2)]
=1/4-1/(n+1)*(n+2)