How would you explain to a student that 1*3*5*...(2n+1) = 1*3*5*...(2n-1)(2n+1)?

Question

anonymous · Answer

Specifically, I have a few students struggling with understanding that
$$\frac{1\cdot3\cdot5 \dotsm(2n+1)}{1\cdot3\cdot5\dotsm(2n-1)}=2n+1.$$

zarkon · Answer

I would write out more of the end terms...

$$\frac{1\cdot3\cdot5 \dotsm(2n+1)}{1\cdot3\cdot5\dotsm(2n-1)}$$

$$=\frac{1\cdot3\cdot5 \dotsm (2n-3)(2n-1)(2n+1)}{1\cdot3\cdot5\dotsm(2n-3)(2n-1)}$$

trojanpoem · Answer

Maybe assume 1.3.5...... until (2n -  3) = k
$$\frac{ k (2n - 1)(2n+1) }{ k(2n-1) }$$ (I am not a teacher)