Chapter 8.4, Problem 38E

Sierpinski's triangle. A geometric figure known as Sierpinski’s triangle is constructed by starting with an equilateral triangle as in the figure. Assume that each side of the triangle has length 1. Inside the original triangle, draw a second triangle by connecting the midpoints of the sides of the original triangle. This divides the original triangle into four identical equilateral triangles. Repeal the process by constructing equilateral triangles inside each of the three smaller blue triangles, again using the midpoints of the sides of the triangle as vertices. Continuing this process with each group of smaller triangles produces Sierpinski's triangle. Coloring each set of smaller triangles helps in following the construction. (Be sure to count blue as well as white triangles.)

1. When a process is repeated over and over, each repetition is called an iteration. How many triangles will the fourth iteration have?
2. How many triangles will the fifth iteration have?
3. Find a formula for the number of triangles in the nth iteration. Use mathematical induction to prove that this formula is correct.