Powers of 3 Modulo 8/Proof 2

Proof
Let the statement be rewritten as:

For all $r \in \Z_{\ge 0}$:
 * $3^r \equiv \begin {cases} 1 \pmod 8 & : r = 2 n \\ 3 \pmod 8 & : r = 2 n + 1 \end {cases}$

where $n \in \Z_{\ge 0}$.

We have:

Then we have:

and the result follows.