# Difference of Two Odd Powers

## 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 $n \to 2 n + 1$.

$\blacksquare$