3. On a line, consider the distinct points $A_{0}, A_{1}, A_{2}, A_{3}, \ldots, A_{2014}$, in this order, such that $M$ is the midpoint of segment $A_{0} A_{10}$ and $A_{0} A_{1}=3 \text{~cm}, A_{1} A_{2}=7 \text{~cm}, A_{2} A_{3}=11 \text{~cm}$, $A_{3} A_{4}=15 \text{~cm}$, and so on.

(3p) a) Calculate the lengths of the segments $\left[A_{9} A_{10}\right]$ and $\left[A_{0} M\right]$.

(4p) b) Calculate the length of the segment $\left[A_{0} A_{2014}\right]$.

Monica Guita
3. a) $A_{0} A_{1}=3=4 \cdot 0+3, A_{1} A_{2}=7=4 \cdot 1+3, A_{2} A_{3}=11=4 \cdot 2+3$,
so $A_{9} A_{10}=4 \cdot 9+3=39$ ..... (1p)
$A_{0} M=A_{0} A_{10}: 2=(3+7+\ldots+39): 2=105$ ..... (2p)
b) $A_{0} A_{2014}=A_{0} A_{1}+A_{1} A_{2}+\ldots+A_{2013} A_{2014}=(4 \cdot 0+3)+(4 \cdot 1+3)+(4 \cdot 2+3)+$
$(4 \cdot 2013+3)=$

(1p)
$=4 \cdot(1+2+\ldots+2013)+2014 \cdot 3=4 \cdot 2013 \cdot 2014: 2+2014 \cdot 3=$ ..... (2p)
$=2014 \cdot 4029=8114406$ ..... (1p)