Question

(1)分解因式：(x+1)(x+2)(x+3)-x(x+1)(x+2) (2)计算：1×2+2×3+3×4+。。。+n（n+1)(n为正整数）

（3）1*2*3+2*3*4+4*5*6+ …+24*25*26，并说明算法

Answer 1

(1)分解因式：(x+1)(x+2)(x+3)-x(x+1)(x+2)
(x+1)(x+2)(x+3)-x(x+1)(x+2)
=(x+1)(x+2)[(x+3)-x]
=3(x+1)(x+2)

(2)计算：1×2+2×3+3×4+……+n(n+1)(n为正整数）
由于n(n+1)=[n(n+1)(n+2)-(n-1)n(n+1)]/3
所以1*2+2*3+……+n(n+1)
=[1*2*3-0+2*3*4-1*2*3+…….+n(n+1)(n+2)-(n-1)n(n+1)]/3
[前后消项]
=[n(n+1)(n+2)]/3
(3)1*2*3+2*3*4+3*4*5+……+24*25*26，并说明算法
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+.....+24*25*26
=24*25*26*27/4
=105300

Answer 2

1)
(x+1)(x+2)(x+3)-x(x+1)(x+2) =(x+1)(x+2)[(x+3)-x]= 3(x+1)(x+2)
2)
1×2+2×3+3×4+。。。+n（n+1)=1+2^2+3^2+...+n^2+1+2+...+n
=n(n+1)(2n+1)/6+n(n+1)/2
=n(n+1)[(2n+1)/6+1/2]
=n(n+1)(3n+2/3)
=n(n+1)(9n+2)/3