Reciprocal of Square of 1 Less than Number Base

Theorem
Let $b \in \Z$ be an integer such that $b > 2$.

Let $n = \left({b - 1}\right)^2$.

The reciprocal of $n$, expressed in base $b$, recurs with period $b - 1$:


 * $\dfrac 1 n = \left[{0.012 \ldots cd012 \ldots}\right]_b$

where:
 * $c = b - 3$
 * $d = b - 1$