2/(1*2*3)+2/(2*3*4)+2/(3*4*5)+…+2/(28*29*30)

问题描述:

2/(1*2*3)+2/(2*3*4)+2/(3*4*5)+…+2/(28*29*30)

2/(1*2*3)+2/(2*3*4)+2/(3*4*5)+…+2/(28*29*30)
=2×( 2/(1×3)-1/2 + 3/(2×4)-1/3 + …… +29/(28×30)-1/29 )
=2×((1+1/3)/2-1/2 + (1/2+1/4)/2-1/3 + …… + (1/28+1/30)/2-1/29 )
=1+1/3+1/2+1/4+……+1/28+1/30 -2×(1/2+1/3+……+1/29)
=1+1/2+2×(1/3+1/4+……+1/28)+1/29+1/30-2×(1/2+1/3+……+1/29)
=1-1/2-1/29+1/30
=8/15-1/29
=217/435