Chapter 7.2, Problem 47MS

White out. Each stage of the Sierpinski triangle is a figure composed of white and green triangles (as depicted on page 471). Mindscape 3 focused on the number of green (filled-in) triangles at each stage, but in this Mindscape we’re going to focus on the white triangles. Counting these bad boys is a little trickier, but you can verify for yourself that at stage n there are ${\text{1+3+3}}^{\text{2}}\text{+}\mathrm{...}{\text{+3}}^{\text{n-1}}$ white triangles. It turns out that this sum can be simplified to equal $\frac{3^{n-1}}{2}.$

Confirm that these two quantities are equal for Stage 4: that is, let $n=4$ and compute the sum and the quotient, verifying that the two answers are equal.