Sum of Sequence of n by 2 to the Power of n/Proof 1

Proof
From Sum of Arithmetic-Geometric Progression:


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

Hence:

Hence the result.