1 . 宋元时期著名数学家朱世杰在其巨著《四元玉鉴》中利用“招差术”得到以下公式:
k(k+1)=
n(n+1)(n+2).具体原理如下:∵k(k+1)=
k(k+1)[(k+2)-(k-1)]=
[k·(k+1)(k+2)-(k-1)k(k+1)],
∴
k(k+1)=
{1×2×3+(2×3×4-1×2×3)+…+[n(n+1)(n+2)-(n-1)n(n+1)]}=
n(n+1)(n+2).
类比上述方法,
k(k+1)(k+2)=____ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76e165f454273d7a4c4db12405861b3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76e165f454273d7a4c4db12405861b3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
类比上述方法,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76e165f454273d7a4c4db12405861b3b.png)
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