Difference between Two Squares equal to Repunit/Corollary 1

Theorem
The sequence of differences of two squares that each make a repunit begins:

and in general for integer $n$:
 * $R_{2 n} = {\underbrace {55 \ldots 56}_{\text {$n - 1$ $5$'s} } }^2 - {\underbrace {44 \ldots 45}_{\text {$n - 1$ $4$'s} } }^2$

that is:
 * $\ds \sum_{k \mathop = 0}^{2 n - 1} 10^k = \paren {\sum_{k \mathop = 1}^{n - 1} 5 \times 10^k + 6}^2 - \paren {\sum_{k \mathop = 1}^{n - 1} 4 \times 10^k + 5}^2$

Proof
From Difference between Two Squares equal to Repunit, $R_{2 n} = x^2 - y^2$ exactly when $R_{2 n} = a b$ where $x = \dfrac {a + b} 2$ and $y = \dfrac {a - b} 2$.

By the Basis Representation Theorem

Thus, let:
 * $a = \paren {10^n + 1}$
 * $b = \ds \sum_{k \mathop = 0}^{n - 1}$

So:

Similarly:

Hence the result.