HCI Sequence and Series Tutorial Q7

The Sierpinski sieve, which is an example of a fractal, is constructed by starting with a solid black equilateral triangle.

This triangle is divided into four congruent equilateral triangles, and the middle triangle is removed (see Figure 1).

On the next step, each of the three remaining equilateral triangles is divided into four congruent equilateral triangles, and the middle triangle in each of these triangles is removed (see Figure 2).If the process is continued indefinitely, the Sierpinski sieve results.

1. Find \({{a}_{k}}\) that gives the number of triangles removed on the \({k^{\mathrm{th}}}\) step.
2. Calculate the number of triangles removed on the fifteenth step.
3. Determine the total area removed after \(12\) steps.
4. Suppose the initial triangle has an area of \(1\) unit square. Find \({{b}_{k}}\) that gives the area removed on the \({k^{\mathrm{th}}}\) step.