I think you're both right. But there are two ways to do what you said and you didn't specify which one.

First, a rectangle of height 1 and width 1/2. The perimeter is 1 * 2 + 1/2 * 2, two sides of height 1 and two sides of width 1/2.

You "glue" the second rectangle. As one may understand this, you glue them by putting them one beside the other standing up, i.e. you glue them along one of the heights. Sorry for the crude ascii art:

    ----   --     ------
    |  |   ||     |    |
    |  |   ||     |    |
    |  | + || ->  |    |
    |  |   ||     |    |
    |  |   ||     |    |
    ----   --     ------

Go on doing that and for a rectangle of width 1/n, you only add 2 * 1/n to the perimeter. In the end you get a single rectangle with height 1 and width 2. The perimeter is 2 * 1 + 2 * 2.

---

Now, maybe, you may want to specify that you glue the rectangles along their widths, not their heights.

That way, the resulting shape when you add the second rectangle is not a rectangle but an irregular shape with 6 sides. Sorry for the crude ascii art again:

         1
    ----------
    |        |n/2
    |        |      1
   n|         ----------
    |                  |n/2
    |                  |
    --------------------
             2

But you didn't specify this option over the other one. And, honestly, if we talk about putting rectangles in a sequence, I think it's more common to think of the rectangles as standing up side by side with their heights together as in the first option. For the second option I would describe the rectangles as having a fixed width of 1 and decreasing heights.