User:Thpigdog/Difference of powers identity

The identity,

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

Proof of the identity,

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

$\ds = \sum_{k \mathop = 0}^{n-1} a^{k+1} b^{n-1-k}-\sum_{k \mathop = 0}^{n-1} a^k b^{n-k} $

$\ds = \sum_{k \mathop = 1}^{n} a^k b^{n-k}-\sum_{k \mathop = 0}^{n-1} a^k b^{n-k} $

$= a^n b^0 - a^0 b^n $

$= a^n - b^n $