<s>[INST] Explaining that $1 \cdot 3 \cdot 5 \dotsm (2n+1) = 1 \cdot 3 \cdot 5 \dotsm (2n-1)(2n+1)$ I have a few students that are having trouble understanding that
$$1 \cdot 3 \cdot 5 \dotsm (2n+1) = 1 \cdot 3 \cdot 5 \dotsm (2n-1)(2n+1),$$
specifically that
$$\frac{1 \cdot 3 \cdot 5 \dotsm (2n+1)}{1 \cdot 3 \cdot 5 \dotsm (2n-1)} = 2n+1.$$
I've tried explaining it a few different ways, but it didn't seem to go over very well.  What would be a good way to explain this?
 [/INST] I would suggest put some various number 
  for example n=3 , and force them to write numbers in two rows 
  then simplify
  $$n=3 \to 2n+1=2(3)+1=7 \to1.3.5.7 \\2n-1=2(3)-1=5 \to 1.3.5\\ \frac{1.3.5...(2n+1)}{1.3.5...(2n-1)}=\frac{1.3.5.7}{1.3.5}$$
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