简便计算1/1*2*3+1/2*3*4+1/3*4*5+.+1/20*21*22

1/1*2*3+1/2*3*4+1/3*4*5+1/4*5*6+1/5*6*7+1/6*7*8+1/7*8*9+1/8*9*10+1/9*10*11=(1/2)[(1/1*2)-(1/2*3)]+(1/2)[(1/2*3)-(1/3*4)]+(1/2)[(1/3*4)-(1/4*5)]+。。。+(1/2)[(1/9*10)-(1/10*11)]=(1/2)[(1/1*2)-(1/2*3)+(1/2*3)-(1/3*4)+(1/3*4)-(1/4*5)+。。。+(1/9*10)-(1/10*11)]=(1/2)[(1/1*2)-(1/10*11)]=(1/2)(1-1/2-1/10+1/11)=(1/2)(2/5+1/11)=(1/2)(27/55)=27/110。