Construct 6 equilateral triangles, the first with a side length of $1 \mathrm{~cm}$ and the following triangles with a side equal to half the side of the previous triangle, as indicated in the figure below. What is the perimeter of this figure?

![](https://cdn.mathpix.com/cropped/2024_05_01_dbce7bc2a200e234c3e2g-09.jpg?height=287&width=417&top_left_y=1315&top_left_x=1236)
The figure is formed by 12 segments, in the sequence of the formation of the triangles.

- 2 segments of $1 \mathrm{~cm}$ and 1 segment of $\frac{1}{2} \mathrm{~cm}$ in triangle I.
- 1 segment of $\frac{1}{2} \mathrm{~cm}$ and 1 segment of $\frac{1}{4} \mathrm{~cm}$ in triangle II.
- 1 segment of $\frac{1}{4} \mathrm{~cm}$ and 1 segment of $\frac{1}{8} \mathrm{~cm}$ in triangle III.
- 1 segment of $\frac{1}{8} \mathrm{~cm}$ and 1 segment of $\frac{1}{16} \mathrm{~cm}$ in triangle IV.
- 1 segment of $\frac{1}{16} \mathrm{~cm}$ and 1 segment of $\frac{1}{32} \mathrm{~cm}$ in triangle V.
- 2 segments of $\frac{1}{32} \mathrm{~cm}$ in triangle VI.

Thus, the perimeter is:

$$
\begin{aligned}
2 \times 1+2 \times \frac{1}{2}+2 \times \frac{1}{4}+2 \times \frac{1}{8}+2 \times \frac{1}{16}+3 \times \frac{1}{32} & =2+1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{3}{32} \\
& =3+\frac{16+8+4+3}{32} \\
& =3+\frac{31}{32}
\end{aligned}
$$