A golden rectangle problem to solve

In my previous blogs on rectangles I showed that the golden rectangle is not so distinctive in its feature that when a square section is removed, the remainder is another rectangle with the same proportions as the first. Similarly, I described a rectangle that when two square sections are removed, the remainder is another rectangle with the same proportions as the first. This can be carried on, infinitely I assume, so that there exist rectangles that when 3, 4, 5, etcetera, square sections are removed, the remainder is another rectangle with the same proportions as the first.

The rectangles I described can be constructed by adding one half unit squares, in series. First you construct a unit square, and then add one half of a unit square on top of the first. Then you draw a line from one corner to the opposite corner of the one and one half unit squares. Then you use that line as the radius to draw an arc that defines the long dimension of the rectangle. To create more rectangles in the series you add one unit square, then one and a half, then two, etcetera.

The problem to solve is to find the formulas (ratios for the long to short sides) for rectangles that are generated in a different series. That is, the golden rectangle is constructed by drawing a line across a half unit square. So, what are the formulas for a series of rectangles constructed by first drawing a line across a third, then a fourth, a fifth, a sixth, etcetera, of a unit square?