Exercise 5

Given an equilateral triangle with sides of length x, we start an iterative process of trisecting its sides and building other – smaller – equilateral triangle on top of the middle part of these trisected sides. These added smaller triangles are then treated in the same fashion: its sides are trisected and smaller equilateral triangles are built on top of the middle part of the trisected sides. See the picture below for an example. From left to right we’ve sketched the basic triangle and the results after the first and second execution of the iterative process.



a)      Denote the perimeter of the resulting figure after the N-th step of the process described above by P(N). Express P(N) in terms of P(N-1) for N > 0. Note that P(0) = 3x.

b)      Use your answer to a) to calculate what happens if you execute the process infinitely many times. Does it converge to a finite value? If yes, calculate this value. If not, explain why not.



If |a|< 1 holds then you may use the following formula