  Let $(x_n)_{n\geq0}$ be a linear recurrence of order $k\geq2$ satisfying
$$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$,
where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In
[`The quotient set of $k$-generalised Fibonacci numbers is dense in
$\mathbb{Q}_p$', \emph{Bull. Aust. Math. Soc.} \textbf{96} (2017), 24-29],
Sanna posed an open question to classify primes $p$ for which the quotient set
of $(x_n)_{n\geq0}$ is dense in $\mathbb{Q}_p$. In this article, we find a
sufficient condition for denseness of the quotient set of the $k$th-order
linear recurrence $(x_n)_{n\geq0}$ satisfying $
x_{n}=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$ for all integers $n\geq k$ with
initial values $x_0=\dots=x_{k-2}=0,x_{k-1}=1$, where $a_1,\dots,a_k\in
\mathbb{Z}$ and $a_k=1$. We show that given a prime $p$, there exist infinitely
many recurrence sequences of order $k\geq 2$ so that their quotient sets are
not dense in $\mathbb{Q}_p$. We also study the quotient sets of linear
recurrence sequences with coefficients in some arithmetic and geometric
progressions.
