Let $S^n$ be the $n$-sphere and consider a $2$-sheeted covering $$ S^n\longrightarrow\mathbb{R}P^n. $$ We have an associated vector bundle $$ \xi: \mathbb{R}^2\longrightarrow S^n\times_{\mathbb{Z}/2}\mathbb{R}^2\longrightarrow \mathbb{R}P^n $$ where the nontrivial element of $\mathbb{Z}/2$ acts on $\mathbb{R}^2$ by reversing the order of coordinates.

**Question:** What is the smallest positive integer $k$ such that the Whitney sum
$
\xi^{\oplus k}
$
is stable equivalent to a trivial bundle?