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

Question

求解，快啊

Answer 1

公式：
n(n+1)(n+2)=(1/4)[n(n+1)(n+2)(n+3)-(n-1)n(n+1)(n+2)]

1×2×3+2×3×4+...+n(n+1)(n+2)
=(1/4)[1×2×3×4-0×1×2×3+2×3×4×5-1×2×3×4+...+n(n+1)(n+2)(n+3)-(n-1)n(n+1)(n+2)]
=n(n+1)(n+2)(n+3)/4

如果不知道这个公式，而知道
1+2+...+n=n(n+1)/2
1²+2²+...+n²=n(n+1)(2n+1)/6
1³+2³+...+n³=[n(n+1)/2]²
的话，也可以展开求和，不过化简的结果还是得到上面的结果。追问

我想知道它的结果，谢谢，能算出来吗？

追答

=n(n+1)(n+2)(n+3)/4
已经写得很清楚了。

Answer 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)

Answer 3

只要知道
1^2+2^2+3^2+…+n^2=n(n+1)(2n+1)/6
1³+2³+...+n³ = (1+2+...+n)² = n²(n+1)²/4
把题目里的式子展开整理就行了

Answer 4

Answer 5

Sn=1*2*3+2*3*4+……+n(n+1)(n+2)
=(1³+2³+3³+----+n³)+3(1²+2²+3²+---+n²)+2(1+2+3+----+n)
=n²(n+1)²/4+n(n+1)(2n+1)/2+n(n+1)
=n(n+1)[n(n+1)/4+(2n+1)/2+1)]
=n(n+1)(n+2)(n+3)/4