1*2*3/1+2*3*4/1+...+9*10*11/1怎样简算?

题目

1*2*3/1+2*3*4/1+...+9*10*11/1怎样简算?
1 1 1
------ + -----+···+ -------等于多少?要简算.
1*2*3 2*3*4 9*10*11

答案

原式=(1/1)*[1/（2*3）]+(1/2)*[1/()3*4]+.+(1/9)[1/(10*11)]
=(1/1)*(1/2-1/3)+(1/2)*(1/3-1/4)+.+(1/9)(1/10-1/11)
=1/(1*2)-1/(1*3)+1/(2*3)-1/(2*4)+.+1/(9*10)-1/(9*11)
=[1/(1*2)+1/(2*3)+.+1/(9*10)]-[1/(1*3）+1/（2*4）+.+1/（9*11）]
=（1-1/2+1/2-1/3+.1/9-1/10）-（1/2）*[2/(1*3）+2/（2*4）+.+2/（9*11）]
=1-1/10-(1/2)*[1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+.+1/8-1/10+1/9-1/11]
=1-1/10-(1/2)*（1+1/2-1/10-1/11）
=1-1/10-1/2-1/4+1/20+1/22
=27/110