Difference of Two Powers/Proof 2

Theorem

Let $\mathbb F$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$.

Let $n \in \N$ such that $n \ge 2$.

Then for all $a, b \in \mathbb F$:

\(\ds a^n - b^n\)

\(=\)

\(\ds \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j\)

\(\ds \)

\(=\)

\(\ds \paren {a - b} \paren {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \dotsb + a b^{n - 2} + b^{n - 1} }\)

Proof

An instance of Difference of Two Powers in a General Commutative Ring.

$\blacksquare$