Reversal formed by Repdigits of Base minus 1 by Addition and Multiplication

Theorem
Let $b \in \Z_{>1}$ be an integer greater than $1$.

Let $n = b^k - 1$ for some integer $k$ such that $k \ge 1$.

Then:
 * $n + n$ is the reversal of $\left({b - 1}\right) n$

when both are expressed in base $b$ representation.

Proof
By Power of Base minus 1 is Repdigit Base minus 1, $n$ is a repdigit number consisting of $k$ occurrences of $b - 1$.

Let $a = b - 1$.

Thus $n$ can be expressed in base $b$ as:
 * $n = {\overbrace {\left[{aaa \cdots a}\right]}^k}_b$

We have that:

where there are:
 * $k - 1$ occurrences of $a$
 * $c = b - 2$.

Then:

where there are:
 * $k - 1$ occurrences of $a$
 * $c = b - 2$.

Hence the result.