3. On a line, consider the points $A_{0}, A_{1}, A_{2}, \ldots, A_{10}$, in this order, such that $A_{1}$ is the midpoint of $\left[A_{0} A_{2}\right]$, $A_{2}$ is the midpoint of $\left[A_{0} A_{3}\right]$, and so on, up to $A_{9}$ being the midpoint of $\left[A_{0} A_{10}\right]$. Knowing that $A_{0} A_{1}=2 \text{~cm}$, determine:

( 3 p ) a ) the length of the segment $A_{9} A_{10}$;

( 4 p ) b ) the length of the side of the equilateral triangle that could be formed from a wire loop with a length equal to $\left(A_{0} A_{10}-1\right) \text{~cm}$.
3. a) $A_{0} A_{1}=A_{1} A_{2}=2 \text{ cm} \Rightarrow A_{0} A_{2}=2 \cdot 2 \text{ cm}=4 \text{ cm}$

$$
A_{2} A_{3}=A_{0} A_{2}=4 \text{ cm} \Rightarrow A_{0} A_{3}=2 \cdot 4 \text{ cm}=8 \text{ cm}=2^{3} \text{ cm}
$$

$A_{3} A_{4}=A_{0} A_{3}=8 \text{ cm} \Rightarrow A_{0} A_{4}=2 \cdot 8 \text{ cm}=16 \text{ cm}=2^{4} \text{ cm} \Rightarrow A_{4} A_{5}=2^{4} \text{ cm}$

$\Rightarrow \cdots A_{9} A_{10}=2^{9} \text{ cm}$
b) $A_{0} A_{10}=A_{0} A_{1}+A_{1} A_{2}+A_{2} A_{3}+\cdots+A_{9} A_{10}=2+2+2^{2}+2^{3}+\cdots+2^{9}=$

$2^{2}+2^{2}+2^{3}+\cdots+2^{9}=2^{3}+2^{3}+\cdots+2^{9}=2^{9}+2^{9}=2^{10}$

$A_{0} A_{10}-1=2^{10}-1=1024-1=1023 \vdots 3$

the length of the sides of the equilateral triangle is 1023:3 $=341 \text{ cm}$