  Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements. For a positive
divisor $r$ of $q^n-1$, the element $\alpha \in \mathbb{F}_{q^n}^*$ is called
\textit{$r$-primitive} if its multiplicative order is $(q^n-1)/r$. Also, for a
non-negative integer $k$, the element $\alpha \in \mathbb{F}_{q^n}$ is
\textit{$k$-normal} over $\mathbb{F}_q$ if $\gcd(\alpha x^{n-1}+ \alpha^q
x^{n-2} + \ldots + \alpha^{q^{n-2}}x + \alpha^{q^{n-1}} , x^n-1)$ in
$\mathbb{F}_{q^n}[x]$ has degree $k$. In this paper we discuss the existence of
elements in arithmetic progressions $\{\alpha, \alpha+\beta, \alpha+2\beta,
\ldots\alpha+(m-1)\beta\} \subset \mathbb{F}_{q^n}$ with $\alpha+(i-1)\beta$
being $r_i$-primitive and at least one of the elements in the arithmetic
progression being $k$-normal over $\mathbb{F}_q$. We obtain asymptotic results
for general $k, r_1, \dots, r_m$ and concrete results when $k = r_i = 2$ for $i
\in \{1, \dots, m\}$.
