We call geometrical identity, or Bernouilli's formula, the following expression:
$$\forall n \in \mathbb{N}, \enspace \forall (a, b) \in \hspace{0.05em} \mathbb{R}^2,$$
$$a^n - b^n = (a-b) \sum_{p=0}^{n-1} a^{n-p-1}b^p $$
Performing the successive Euclidian divisions of the polynomial \( (a^n - b^n) \) by \( (a - b) \), we notice that:
And finally,
$$\forall n \in \mathbb{N}, \enspace \forall (a, b) \in \hspace{0.05em} \mathbb{R}^2,$$
$$a^n - b^n = (a-b) \sum_{p=0}^{n-1} a^{n-p-1}b^p $$
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