Question

1*2*3+2*3*4+3*4*5+4*5*6+5*6*7+6*7*8+7*8*9+8*9*10=

Answer 1

因为4(n-1)n(n+1)=(n-1)n(n+1)(n+2)-(n-2)(n-1)n(n+1)
所以
1*2*3+2*3*4+3*4*5+......+n(n+1)(n+2)
=n(n+1)(n+2)(n+3)÷4

1*2*3+2*3*4+3*4*5+4*5*6+5*6*7+6*7*8+7*8*9+8*9*10
=8×9×10×11/4
=1980

Answer 2

原式=6[C(3,3)+C(4,3)+C(5,3)+C(6,3)+C(7,3)+C(8,3)+C(9,3)+C(10,3)]=6[C(4,4)+C(4,3)+C(5,3)+C(6,3)+C(7,3)+C(8,3)+C(9,3)+C(10,3)]=6[C(5,4)+C(5,3)+C(6,3)+C(7,3)+C(8,3)+C(9,3)+C(10,3)]=…=6C(10,4)=10*9*8*7/4=1260。（C(m,n)是组合数）

Answer 3

1*2*3+2*3*4+3*4*5+4*5*6+5*6*7+6*7*8+7*8*9+8*9*10
=(1*2*3*4-0*1*2*3)/4+(2*3*4*5-1*2*3*4)/4+...+(8*9*10*11-7*8*9*10)/4
=8*9*10*11/4
=1980

Answer 4

∑(n-1)n(n+1),n从2到9
=∑(n^3-n)
=∑n^3-∑n
=[(9*10/2)^2-1]-[(2+9)*8/2]
=1980
立方和公式:1^3 + 2^3 +…… n^3 = [n (n+1) / 2] ^2