Let $n$ be a non-zero natural number. We set
$$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$
where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$.
Show that $X_n$ and $-X_n$ have the same distribution for all $n \in \mathbb{N}^{\star}$.