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?



covering map from spheres to projective spaces and the associated vector bundle
Theorem 7.4 of J. F. Adams, Vector fields on spheres, Ann. of Math. 75 (1962), 603–632
says that $$\tilde{KO}({\mathbb R} P^n)=\mathbb Z\,/\,2^{\phi(n)},$$ generated by $\xi-1$, where $\phi(n)$ is the number of integers $s$ such that $0 < s\le n$ and $s$ is congruent to 0,1, 2 or 4 modulo 8, and $\xi$ is the canonical line bundle over $\mathbb RP^n$ given as $S^n\times\_{\mathbb Z/2} \mathbb R\to \mathbb RP^n$, where $\mathbb Z/2$ acts on $ \mathbb R$ by multiplication with $-1$.


Since your vector bundle is the direct sum of $\xi$ and the trivial 1-dimensional bundle, the answer to your question is $k=2^{\phi(n)}$.


