I pick up a stitch every other row, so the first row I pick up a stitch I have knit stitch at the end (so, picking up the stitch at the end makes that stitch p). The next time I pick up a stitch, the last stitch is p (picked up stitch is k).
When you say that you “pick up at stitch” do you mean you add a stitch (increase)?
If you do it like Sue said just be sure when you get past where you did the edge stitch and increased one that you knit the knits and purl the purls, then you won’t lose your ribbing.
I have added stitches right at the end in ribbing by working front back into the edge stitch. For instance if the edge stitch is knit, I’d knit it in the front loop and purl it in the back loop, then turn and k1, p1, etc.
If the last stitch was a purl I’d purl it through the front loop and then knit it in the back loop. Turn and p1, k1, etc.