What is the best way to denote it?
- $\forall i\in\{1,\dots,n\}: P(i)$;
- $\forall i=1,\dots,n: P(i)$;
- $\forall i=\overline{1,n}: P(i)$;
- $P(i)$ for $i=1,\dots,n$;
- ...
What is the best way to denote it?
Use "$P(i)$ for $i=1,...,n$" to refer to the property being true in those cases. Using existential and universal quantifiers in ordinary mathematical prose (as opposed to formal logic text) is ugly.
I don't know if I've ever seen the notation $\overline{1, n}$, so I wouldn't use that :) But, maybe others know it, so maybe it's okay. Other than that, I wouldn't use $\forall$ or $:$ ever and most of my experience with such notation is in the class where you first learn it. I would use words but the basic forms of your 1, 2, and 4 seem pretty good.
$P(i)$ is true for all $i$ such that $1 \leq i \leq n$.
$P(i)$ is true for $i = 1, \ldots, n$.
For $i = 1, \ldots, n$, $P(i)$ is true.