Question

1*2*3分之1+2*3*4分之1+3*4*5分之1+……+20*21*22分之1 简便计算

1*2*3分之1+2*3*4分之1+3*4*5分之1+……+20*21*22分之1
简便计算

Answer 1

1/(1×2×3)=1/2[1/(1×2)-1/(2×3)]
1/(2×3×4)=1/2[1/(2×3)-1/(3×4)]
...
1/(20×21×22)=1/2[1/(20×21)-1/(21×22)]
左右两边分别相加，中间项都是一正一负消去了，只剩首尾项
1/2[1/(1×2)-1/(21×22)]=1/4[1-1/21×11]=115/462
即1/(1×2×3)+1/(2×3×4)+······+1/(20×21×22)=115/462

Answer 2

Answer 3

=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×(1/20×21-1/21×22)
=1/2×(1/1×2-1/21×22)
=115/231

追答

=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×(1/20×21-1/21×22)
=1/2×(1/1×2-1/21×22)
=115/462
【前面计算错了，特此更正】