Exercise 12.

Starting with any rectangle, we can create a new, larger rectangle by attaching a square to the longer side. For example, if we start with a \(2\times 5\) rectangle, we would glue on a \(5\times 5\) square, forming a \(5 \times 7\) rectangle:

On the left, a 5x5 square to the right of a rectangle with base 2 and height 5, separated by a small gap.  An arrow points to the right, where a rectangle of base 7 and height 5 is shown, including a dotted line representing where the square and triangle on the left were glued together.

The next rectangle would be formed by attaching a \(7 \times 7\) square to the top or bottom of the \(5\times 7\) rectangle.

  1. Create a sequence of rectangles using this rule starting with a \(1\times 2\) rectangle. Then write out the sequence of perimeters for the rectangles (the first term of the sequence would be 6, since the perimeter of a \(1\times 2\) rectangle is 6 - the next term would be 10).

  2. Repeat the above part this time starting with a \(1 \times 3\) rectangle.

  3. Find recursive formulas for each of the sequences of perimeters you found in parts (a) and (b). Don't forget to give the initial conditions as well.

  4. Are the sequences arithmetic? Geometric? If not, are they close to being either of these (i.e., are the differences or ratios almost constant)? Explain.

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