Assumption: $(n+1)(n+2) \cdots (2n) = (2^n)\cdot 1 \cdot 3 \cdot 5 \cdots (2n-1)$
Prove for $n+1$:
$(n+2)(n+3) \cdots (2(n+1)) = (2^{n+1}) \cdot 1 \cdot 3 \cdot 5 \cdots (2(n+1)-1)$
Using the assumption, I divide both sides by $(n+1)$ and substitute RHS into my $n+1$ equation, however it does not equate.