Smaller and Smaller

Caroline and Bronya from Tattingstone School sent wonderfully explained solutions to this problem. Caroline drew some diagrams to help her:

	Side AB = 1 Perimeter = 3 x 1 = 3
	Side ADB = 4 x 1/3 = 4/3 Perimeter = 3 x 4/3 = 4
	Side ADB = 4 x 4/9 = 16/9 Perimeter = 3 x 16/9 = 48/9 = 16/3 = 5 1/3

She explains:

The pattern is every time you add on a new smaller spike you multiply by 4 and divide by 3.

Here, Bronya describes how she approached the problem:

First of all I looked at the perimeter of the equilateral triangle. It was 3 units. Then I looked at the perimeter of the star. If one side is 1¹/₃ units then the perimeter is 4 units. Then the perimeter of the third shape. I looked at a section like this:

Each section = ¹/₃ + ¹/₉ unit = ³/₉ + ¹/₉ = ⁴/₉ unit
There are 12 sections so the total perimeter = ⁴/₉ x ¹²/₁ = ⁴⁸/₉
I looked at all the perimeters as ninths:
Perimeter 1 was ²⁷/₉
Perimeter 2 was ³⁶/₉
Perimeter 3 was ⁴⁸/₉
Each time the perimeter increases by one third.
I think this comes about because in each section a third of the section is added on.

The perimeter of the next shape would be ⁶⁴/₉ because:
48 divided by 3 is 16 therefore increase ⁴⁸/₉ by ¹⁶/₉
Total would be: ⁴⁸/₉ + ¹⁶/₉ = ⁶⁴/₉

Thank you to you both - these are very well reasoned solutions. I wonder whether you could generalise to any shape in this series?