Difference of Two Odd Powers

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Theorem

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

Let $n \in \Z_{\ge 0}$ be a positive integer.


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

\(\displaystyle a^{2 n + 1} - b^{2 n + 1}\) \(=\) \(\displaystyle \paren {a - b} \sum_{j \mathop = 0}^{2 n} a^{2 n - j} b^j\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {a - b} \paren {a^{2 n} + a^{2 n - 1} b + a^{2 n - 2} b^2 + \dotsb + a b^{2 n - 1} + b^{2 n} }\)


Proof

A direct application of Difference of Two Powers:

\(\displaystyle a^n - b^n\) \(=\) \(\displaystyle \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {a - b} \paren {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \dotsb + a b^{n - 2} + b^{n - 1} }\)

and setting $n \to 2 n + 1$.

$\blacksquare$


Also see


Sources