\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Show that the family \(\bigl(1 + X, X + X^2, X^2 + X^3, \ldots, X^{n-1} + X^n, X^n\bigr)\) is a basis of \(\mathbb{R}_n[X]\) for every \(n \in \mathbb{N}^*\).
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