連乘積和

Processing math: 100%

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