Question

小学计算题

1/(1*2*3)+1/(2*3*4)+1/(3*4*5)+...........1/(20*21*22)

Answer 1

1/n(n+1)(n+2)=[1/n(n+1)-1/(n+1)(n+2)]*1/2
1/（1×2×3）+ 1/（2×3×4）+1/（3×4×5）+……+ 1/（20×21×22）
=(1/1*2-1/2*3+1/2*3-1/3*4+1/3*4-1/4*5+……+1/20*21-1/21*22)*1/2
=(1/2-1/21*22)*1/2
=115/462

Answer 2

1/n(n+1)(n+2)=[1/n(n+1)-1/(n+1)(n+2)]*1/2
1/（1×2×3）+ 1/（2×3×4）+1/（3×4×5）+……+ 1/（20×21×22）
=(1/1*2-1/2*3+1/2*3-1/3*4+1/3*4-1/4*5+……+1/20*21-1/21*22)*1/2
=(1/2-1/21*22)*1/2
=115/462 知道了吧