求数列1/2*5，1/5*8，1/8*11，...，的前n项和Sn

1/2*5，1/5*8，1/8*11，...，
分母公差是3.
1/2*5=1/3*[1/2-1/5]
1/5*8=1/3*[1/5-1/8]
.....
1/(3n-1)(3n+2)=1/3*[1/(3n-1)-1/(3n+2)]
Sn=1/3*[1/2-1/5+1/5-1/8+1/8-1/11+....+1/(3n-1)-1/(3n+2)]
=1/3*[1/2-1/(3n+2)]
=1/3*[(3n+2-2)/2(3n+2)]
=n/2(3n+2)

3/2*5=(5-2)/2*5=5/2*5-2/2*5=1/2-1/5
3/5*8=(8-5)/5*8=8/5*8-5/5*8=1/5-1/8
以此类推

1/2*5+1/5*8+……+1/(2n-1)(2n+2)
=[3/2*5+3/5*8+……+3/(2n-1)(2n+2)]/3
=[(1/2-1/5)+(1/5-1/8)+……+1/(2n-1)-1/(2n+2)]/3
=[1/2-1/(2n+2)]/3
=n/(6n+6)