Sum of Sequence of Power by Index/Proof 2

Proof
From Sum of Arithmetic-Geometric Progression:


 * $\displaystyle \sum_{j \mathop = 0}^n \left({a + j d}\right) x^j = \frac {a \left({1 - x^{n + 1} }\right)} {1 - x} + \frac {x d \left({1 - \left({n + 1}\right) x^n + n x^{n + 1} }\right)} {\left({1 - x}\right)^2}$

Hence:

Hence the result.