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

Proof
From Sum of Arithmetic-Geometric Sequence:


 * $\ds \sum_{j \mathop = 0}^n \paren {a + j d} r^j = \frac {a \paren {1 - r^{n + 1} } } {1 - r} + \frac {r d \paren {1 - \paren {n + 1} r^n + n r^{n + 1} } } {\paren {1 - r}^2}$

Hence:

Hence the result.