1乘2乘3+2乘3乘4+3乘4乘5 +.....+N乘(N+1)乘(N+2)=

1×2×3+2×3×4+3×4×5+......+n(n+1)(n+2) =1/4【1×2×3×4-0×1×2×3】+1/4【2×3×4×5-1×2×3×4】+1/4【3×4×5×6-2×3×4×5】+......+ 1/4【n(n+1)(n+2)（n+3)-(n-1)n(n+1)(n+2)】 =1/4n(n+1)(n+2)(n+3)

1×2×3+2×3×4+3×4×5+......+n(n+1)(n+2) =4×1/4×【1×2×3+2×3×4+3×4×5+......+n(n+1)(n+2)】=1/4×【1×2×3×4+2×3×4×4+3×4×5×4+......+n(n+1)(n+2）×4】=1/4×｛1×2×3×（4－0）+2×3×4（5－1）+3×4×5（6－2）+......+n(n+1)(n+2)【（n＋3）-（n-1）】｝＝1/4×【1×2×3×（4－0）】+1/4×【2×3×4（5－1）】+1/4×【3×4×5（6－2）】+......+1/4×｛n(n+1)(n+2)【（n＋3）-（n-1）】｝＝1/4【1×2×3×4-0×1×2×3】+1/4【2×3×4×5-1×2×3×4】+1/4【3×4×5×6-2×3×4×5】+......+ 1/4【n(n+1)(n+2)（n+3)-(n-1)n(n+1)(n+2)】 =1/4n(n+1)(n+2)(n+3)

6+24+60+120+210+336
18 36 60 90 126
18 24 30 36
+6 +6 +6
所以，它等于18+6N

=1/2×(1/1×2-1/2×3+1/2×3-1/3×4+1/3×4-1/4×5)=1/2×(1/1×2-1/4×5)=9/40