Repunit Integer as Product of Base - 1 by Increasing Digit Integer/General Result

Theorem
Let $b \in \Z_{>1}$.

Then:
 * $\ds \paren {b - 1} \sum_{j \mathop = 0}^n \paren {n - j} b^j + n + 1 = \sum_{j \mathop = 0}^n b^j$