Question

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

Answer 1

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

Answer 2

I would write out more of the end terms...

\[\frac{1\cdot3\cdot5 \dotsm(2n+1)}{1\cdot3\cdot5\dotsm(2n-1)}\]

\[=\frac{1\cdot3\cdot5 \dotsm (2n-3)(2n-1)(2n+1)}{1\cdot3\cdot5\dotsm(2n-3)(2n-1)}\]

Answer 3

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