User:Thpigdog/Difference of powers identity
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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 $