Question

)在计算“1×2+2×3+…+n(n+1)”时,某同学学到了如下一种方法:先改写第k项:

k(k+1)=[k(k+1)(k+2)-(k-1)k(k+1)],

由此得1×2=(1×2×3-0×1×2),

2×3=(2×3×4-1×2×3),…,

n(n+1)=[n(n+1)(n+2)-(n-1)n(n+1)].

相加,得1×2+2×3+…+n(n+1)=n(n+1)(n+2).

类比上述方法,请你计算“1×2×3+2×3×4+…+n(n+1)(n+2)”,其结果为　　　　.

 

Answer 1

【答案】

n(n+1)(n+2)(n+3)

【解析】k(k+1)(k+2)=[k(k+1)(k+2)(k+3)-(k-1)k(k+1)(k+2)],

∴1×2×3+2×3×4+…+n(n+1)(n+2)=n(n+1)(n+2)(n+3).