1/1*2*3=(1/1-1/2-1/2+1/3)/2
1/2*3*4=(1/2-1/3-1/3+1/4)/2
1/3*4*5=(1/3-1/4-1/4+1/5)/2
.
.
.
1/20*21*22=(1/20-1/21-1/21+1/22)/2
1/1*2*3+1/2*3*4+...+1/20*21*22
=(1/1-1/2-1/2+1/3+1/2-1/3-1/3+1/4+1/3-1/4-1/4+1/5+.+1/18-1/19-1/19+1/20+1/19-1/20-1/20+1/21+1/20-1/21-1/21+1/22)/2
=(1-1/2-1/21+1/22)/2
=115/462