連乘積和

連續乘積和的等式

n是正整數

$(n+1)\displaystyle\prod_{x=1}^{n}x$=$\displaystyle\prod_{x=1}^{n+1}x$−$\displaystyle\prod_{x=0}^{n}x$

$(n+1)\displaystyle\prod_{x=2}^{n+1}x$=$\displaystyle\prod_{x=2}^{n+2}x$−$\displaystyle\prod_{x=1}^{n+1}x$

$(n+1)\displaystyle\prod_{x=3}^{n+2}x$=$\displaystyle\prod_{x=3}^{n+3}x$−$\displaystyle\prod_{x=2}^{n+2}x$

$(n+1)\displaystyle\prod_{x=4}^{n+3}x$=$\displaystyle\prod_{x=4}^{n+4}x$−$\displaystyle\prod_{x=3}^{n+3}x$

.....................

$(n+1)\displaystyle\prod_{x=n}^{n+n-1}x$=$\displaystyle\prod_{x=n}^{n+n}x$−$\displaystyle\prod_{x=n-1}^{n+n-1}x$

$(n+1)\displaystyle\prod_{x=n+1}^{n+n}x$=$\displaystyle\prod_{x=n+1}^{n+n+1}x$−$\displaystyle\prod_{x=n}^{n+n}x$

     將左列各式加總得

     $(n+1)\displaystyle\sum_{i=1}^{n+1}\prod_{x=i}^{n+i-1}x$=$\displaystyle\prod_{x=n+1}^{n+n+1}x$−$\displaystyle\prod_{x=0}^{n}x$=$\displaystyle\prod_{x=n+1}^{n+n+1}x$

     所以

     $\displaystyle\sum_{i=1}^{n+1}\prod_{x=i}^{n+i-1}x$=$\displaystyle\prod_{x=n+2}^{2n+1}x$

     例如

    n=1，$\displaystyle\sum_{i=1}^{2}\prod_{x=i}^{i}x$=$\displaystyle\prod_{x=3}^{3}x$，即 1+2＝3

   n=2，$\displaystyle\sum_{i=1}^{3}\prod_{x=i}^{i+1}x$=$\displaystyle\prod_{x=4}^{5}x$，即 1×2+ 2×3+ 3×4＝ 4×5

   n=3，$\displaystyle\sum_{i=1}^{4}\prod_{x=i}^{i+2}x$=$\displaystyle\prod_{x=5}^{7}x$，即 1×2×3+ 2×3×4+ 3×4×5+4×5×6＝ 5×6×7

      n=4，$\displaystyle\sum_{i=1}^{5}\prod_{x=i}^{i+3}x$=$\displaystyle\prod_{x=6}^{9}x$，即
      1×2×3×4+ 2×3×4×5+3×4×5×6+4×5×6×7+5×6×7×8＝ 6×7×8×9

[註1]：$\displaystyle\sum_{i=1}^{5}$=1+2+3+4+5
[註2]：$\displaystyle\prod_{i=1}^{5}$=1×2×3×4×5