Exercise 75 Page 281

As we can see, $\text{six}$ cycles of the $\text{three}\text{-}$color pattern will be completed. Moreover, the twentieth figure will have the $\text{second}$ color of the $\text{three}\text{-}$color pattern. Therefore, the twentieth figure will be blue.

As we can see, $\text{seven}$ cycles of $\text{three}$ figures will be completed. Then, the $23^{\text{rd}}$ figure will be the $\text{second}$ figure of the eighth cycle. If we take each cycle as a term, we can write an arithmetic sequence. Let's recall the explicit formula for an arithmetic sequence. Here, $A(1)$ is the first term and $d$ is the common difference. In this case, each cycle is one term. In the first cycle of figures, we have triangles — $3$ sides. Then, we have squares — $4$ sides. This pattern continues, adding $1$ to the number of sides each time. Therefore, the common difference is $1.$ With this information, we can write the formula. Let's evaluate this formula for the eighth cycle of figures, where the $23^{\text{rd}}$ figure belongs. Therefore, the $23^{\text{rd}}$ figure will have $10$ sides.