I have a problem. I worked on it for some time and I've got this: $$\frac{\sum_{k=0}^{n/2} v^{2k}}{\sum_{k=0}^{n/2} v^{2k+1}}=\frac{1}{2}$$ The only thing we know about $n$ is that it is even.
Is it possible to infer $v$ from this? If not, I guess I'll have to approach the problem in different way.
Since$$\frac{\sum_{k=0}^{n/2} v^{2k}}{\sum_{k=0}^{n/2} v^{2k+1}}=\frac{\sum_{k=0}^{n/2} v^{2k}}{v\sum_{k=0}^{n/2} v^{2k}}=\frac1v,$$you can infer that $v=2$.