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 can’t see the images — could you upload them or describe how each image changes from the previous one (what is added or removed each step)?

Meanwhile, here’s how to find the recursive formula and common possibilities:

1. Compute the first few perimeters P1, P2, P3… from the pictures.
2. Look at the change between consecutive terms: Δn = Pn − P{n−1}.
- If the change is constant (Δn = d), the recursion is linear-additive:
an = a{n−1} + d, with a1 = (first perimeter).
Example: if each image adds 6 units to the perimeter, an = a{n−1} + 6, a1 = (value).
- If each perimeter is a constant multiple of the previous (ratio r), the recursion is multiplicative:
an = r · a{n−1}, with a1 = (first perimeter).
Example: if each perimeter doubles, an = 2·a{n−1}, a1 = (value).

3. If the change itself follows a pattern (e.g., increases by a fixed amount each time), you may need a second-order recursion:
an = an−1 + (bn−1) or an = 2·an−1 − an−2, etc.

If you upload the images or give P1, P2, P3 (or describe what’s added each step), I’ll give the exact recursive formula and the initial condition.