$$\lim_{n \to \infty}\left(\prod_{r=0}^{n/2}{\left(\frac{mn+1-mr}{mn+3-mr}\right)}\right)$$
Of course if we check out cases where it should telescope like $ m = 1, 2, \frac{2}{3}, \cdots$, we get closed forms - $\frac{1}{4}, \frac{1}{2}, \frac{1}{8}, \cdots$.
So, I was thinking if it has a closed form.
One can reduce it to
$\displaystyle \lim_{n \to \infty} \dfrac{\prod_{r=0}^n{\frac{r + p/m}{r+q/m}}}{\prod_{r=0}^{n/2}{\frac{r + p/m}{r+q/m}}}$, where $p,q$ are $1,3$ in the above problem
$$\displaystyle = \lim_{n \to \infty}{\dfrac{\Gamma\left(\frac{p}{m}+n+1\right) \Gamma\left(\frac{q}{m} + \frac{n}{2} + 1\right)}{\Gamma\left(\frac{q}{m}+n+1\right) \Gamma\left(\frac{p}{m} + \frac{n}{2} + 1\right)}}$$
Now, how to find this limit? Wolfram Alpha now can find the limit quite fast for example for $(p,q) = (1,3)$, and $m=3$ it gives the limit $\frac{\sqrt[3]{2}}{2}$.