1/(1*2*3)+1/(2*3*4)+...+1/(20*21*22)
=1/2(1/1*2-1/2*3)+1/2(1/2*3-1/3*4)+...+1/2(1/20*21-1/21*22)
=1/2(1/1*2-1/2*3+1/2*3-1/3*4+...+1/20*21-1/21*22)
=1/2(1/1*2-1/21*22)
=1/2*115/231
=115/462
1/(1*2*3)+1/(2*3*4)+...+1/(20*21*22)
=1/2(1/1*2-1/2*3)+1/2(1/2*3-1/3*4)+...+1/2(1/20*21-1/21*22)
=1/2(1/1*2-1/2*3+1/2*3-1/3*4+...+1/20*21-1/21*22)
=1/2(1/1*2-1/21*22)
=1/2*115/231
=115/462