Question

Which recursive formula describes the pattern in the perimeters of the images, where an is the perimeter of the nth image?

Answer 1

I don't see the images — can you upload them or list the first few perimeters? Without the pictures, I can only give the method and the common forms:

How to find the recursive formula
1. Measure (or read) a1, a2, a3 (the perimeters of images 1, 2, 3).
2. Check the differences d = a2 − a1, a3 − a2:
- If the differences are the same, the sequence is arithmetic and the recursion is
an = an−1 + d (with given a1).
3. If not arithmetic, check ratios r = a2 / a1, a3 / a2:
- If the ratios are the same, the sequence is geometric and the recursion is
an = r · an−1 (with given a1).
4. If neither constant difference nor constant ratio, look for a pattern from the construction (e.g., each step adds k copies of some boundary length), and write an = an−1 + (increment expressed in n or in an−1).

Common examples
- If each image doubles the perimeter: an = 2·an−1 (geometric).
- If each image adds 6 units of length: an = an−1 + 6 (arithmetic).
- If each step multiplies by 3/2: an = (3/2)·an−1.

If you paste the first 2–3 perimeter values or upload the images, I’ll give the exact recursive formula for this pattern.