difference of two nth powers (proof)

$\displaystyle a^1 - b^1 = (a - b)$

$\displaystyle a^2 - b^2 = (a-b)(a+b)$

$\displaystyle a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

$\displaystyle a^4 - b^4 = (a-b)(a^3 + a^2b + ab^2 + b^3)$

If you continue with increasing the exponent then you notice the following general rule: $\displaystyle a^n - b^n = (a-b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + ... + ab^{n-2} + b^{n-1})$

However, this doesn't proof that this rule is valid for every positive integer exponent. How can you proof that this rule indeed is valid for every positive integer exponent?