Fickle predicate that is eventually always true
Let $P(n)$ be "$n$ is either odd or the product of two distinct numbers more than $1$.".
The sequence of truth-values of $P$ on $1,2,3,...$ is:
$1,0,1,0,1,1,1,...$
It can of course be proven by induction that $P(n)$ is true for every integer $n > 4$.
Predicate that alternates for a long time
Just for fun, here is a predicate that alternates for $9$ terms before breaking the pattern on the $10$-th. However, it does not have the required property of being true of sufficiently large positive integers.
Let $Q(n)$ be "The minimum number of unit-length matchsticks needed to form $n$ unit equilateral triangles is either $2$ or $3$ more than a multiple of $4$.".
The sequence of minimum number of matchsticks needed is:
$3,5,7,9,11,12,14,16,18,19,21,23,24,...$
The sequence of truth-values of $P$ on $1,2,3,...$ is:
$1,0,1,0,1,0,1,0,1,1,0,1,0,...$