  Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$
represents a finite extension of degree $n$ of the finite field
${\mathbb{F}_{q}}$. In this article, we investigate the existence of $m$
elements in arithmetic progression, where every element is primitive and at
least one is normal with prescribed norms. Moreover, for $n\geq6,q=3^k,m=2$ we
establish that there are only $10$ possible exceptions.
