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Question Number 182584 by Matica last updated on 11/Dec/22

  Transform into factorial form the following    A = 1×3×5×7×9×...×(2n−1) .   Example: 4×3×2×1=4!

$$\:\:{Transform}\:{into}\:{factorial}\:{form}\:{the}\:{following} \\ $$$$\:\:{A}\:=\:\mathrm{1}×\mathrm{3}×\mathrm{5}×\mathrm{7}×\mathrm{9}×...×\left(\mathrm{2}{n}−\mathrm{1}\right)\:.\: \\ $$$${Example}:\:\mathrm{4}×\mathrm{3}×\mathrm{2}×\mathrm{1}=\mathrm{4}! \\ $$

Answered by aleks041103 last updated on 11/Dec/22

A=1.3.5.7...(2n−1)=((1.2.3.4.5.6.7...(2n−1).2n)/(2.4.6....(2n)))=  =(((2n)!)/((2.1).(2.2).(2.3)....(2.n)))=(((2n)!)/(2^n n!))  ⇒A=(((2n)!)/(2^n n!))

$${A}=\mathrm{1}.\mathrm{3}.\mathrm{5}.\mathrm{7}...\left(\mathrm{2}{n}−\mathrm{1}\right)=\frac{\mathrm{1}.\mathrm{2}.\mathrm{3}.\mathrm{4}.\mathrm{5}.\mathrm{6}.\mathrm{7}...\left(\mathrm{2}{n}−\mathrm{1}\right).\mathrm{2}{n}}{\mathrm{2}.\mathrm{4}.\mathrm{6}....\left(\mathrm{2}{n}\right)}= \\ $$$$=\frac{\left(\mathrm{2}{n}\right)!}{\left(\mathrm{2}.\mathrm{1}\right).\left(\mathrm{2}.\mathrm{2}\right).\left(\mathrm{2}.\mathrm{3}\right)....\left(\mathrm{2}.{n}\right)}=\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}^{{n}} {n}!} \\ $$$$\Rightarrow{A}=\frac{\left(\mathrm{2}{n}\right)!}{\mathrm{2}^{{n}} {n}!} \\ $$

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