1/1*3 1/2*4 1/3*5 1/4*6 ...1/18*20

1/n(n+2)=1/2[1/n-1/(n+2)]
1/1*3+1/2*4+1/3*5+1/4*6+...+1/18*20
=1/2(1-1/3+1/2-1/4+1/3-1/5+……+1/17-1/19+1/18-1/20)
=1/2(1+1/2-1/19-1/20)
=1/2(3/2-39/380)
=1/2*531/380
=531/760

1/1*3+1/2*4+1/3*5+1/4*6+...+1/18*20
=1/2*[(1/1-1/3)+(1/2-1/4)+(1/3-1/5)+(1/4-1/6)+....+(1/16-1/18)+(1/17-1/19)+(1/18-1/20)]
=1/2*(1+1/2-1/19-1/20)
=1/2*29/20
=1/2*531/380
=531/760

1/1*3+1/2*4+1/3*5+1/4*6+...+1/18*20
=(1/1-1/3)/2+(1/2-1/4)/2+(1/3-1/5)/2+(1/4-1/6)/2+......+(1/18+1/20)/2
=(1/1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+....+1/18-1/20)/2
=(1+1/2-1/19-1/20)/2
=(531/380)/2
=531/760

1/1*3=1/2(1-1/3) 吧所有项都展开成这样
可得 1+1/2-1/19-1/20=531/380

原式=1/2*(1-1/3)+1/2*(1/2-1/4)+...+1/2(1/18-1/20)
=1/2*(1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+.....+1/16-1/18+1/17-1/19+1/18-1/20 )
=1/2*(1+1/2-1/19-1/20)
=531/760