Deduce that there exists $\alpha \in \mathbb{C}$ such that, for all $k$ in $\llbracket 0, n+1 \rrbracket$, $x_k = 2\mathrm{i}\alpha \frac{\rho^k}{b^k} \sin\left(\frac{\ell k \pi}{n+1}\right)$.
Deduce that there exists $\alpha \in \mathbb{C}$ such that, for all $k$ in $\llbracket 0, n+1 \rrbracket$, $x_k = 2\mathrm{i}\alpha \frac{\rho^k}{b^k} \sin\left(\frac{\ell k \pi}{n+1}\right)$.