Question

1x2×3十2x3x4十3x4X5+…十9X10x11的值是的少

1x2×3十2x3x4十3x4X5+…十9X10x11的值是的少

Answer 1

1×2＋2×3＋3×4＋...＋n×(n+1)

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

= (1/3)×n×(n+1)×(n+2)

[2] 相邻三个正整数乘积和

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)]

= (1/4)×n×(n+1)×(n+2)×(n+3)

[3] 相邻m个正整数乘积和

＜k=1 to n＞∑k×(k+1)×(k+2)×...×(k+m-1

... 展开

∑〈1，n〉n(n+1)(n+2)
=∑〈1，n〉n^3+3∑〈1，n〉n^2+2∑〈1，n〉n
=[n(n+1)/2]^2+3[n(n+1)(2n+1)/6]+2[n(n+1)/2]
=[n(n+1)/2][(n^2+n)/2+(2n+1)+2]
=(1/4)n(n+1)(n^2+5n+6)
=n(n+1)(n+2)(n+3)/4
即一般的求和公式为：
S=n(n+1)(n+2)(n+3)/4.
∴n=10时，代入以上求和公式，得
1·2·3+2·3·4+……+10·11·12
=(10·11·12·13)/4
=390·11