User:Thpigdog/Difference of powers identity

The identiy,
 * $ \displaystyle a^n - b^n = (a-b) \sum_{k=0}^{n-1} a^k b^{n-1-k} $

Proof of the identity,


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


 * $ \displaystyle = \sum_{k=0}^{n-1} a^{k+1} b^{n-1-k}-\sum_{k=0}^{n-1} a^k b^{n-k} $


 * $ \displaystyle = \sum_{k=1}^{n} a^k b^{n-k}-\sum_{k=0}^{n-1} a^k b^{n-k} $


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


 * $ = a^n - b^n $