2. On a line, points $A_{1}, A_{2}, \ldots, A_{20}$ are taken in this order, such that $A_{1} A_{2}=6 \text{~cm}$, $A_{2} A_{3}=12 \text{~cm}$, $A_{3} A_{4}=18 \text{~cm}$, and so on.

a) What is the length of the segment $\left[A_{1} A_{20}\right]$? What about the segment $\left[A_{15} A_{20}\right]$?

b) Determine $i \in \mathbb{N}^{*}$ for which $M \in\left[A_{i} A_{i+1}\right]$, where $M$ is the midpoint of $\left[A_{1} A_{20}\right]$.
2. a) $A_{1} A_{20}=6 \cdot(1+2+\ldots+19)=1140 \mathrm{~cm}$, and $A_{15} A_{20}=6 \cdot(15+16+\ldots+19)=510 \mathrm{~cm}$.

b) According to point a), we have that $A_{1} M=570 \mathrm{~cm}$; then

$$
M \in\left[A_{i} A_{i+1}\right] \Leftrightarrow A_{1} A_{i} \leq A_{1} M \leq A_{1} A_{i+1} \Leftrightarrow 6 \cdot \frac{(i-1) i}{2} \leq 570 \leq 6 \cdot \frac{i(i+1)}{2}
$$

and this happens only for $i=14$.