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$

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


 

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