3. (7p) On a line, consider the distinct points $A_{0}, A_{1}, A_{2}, \ldots, A_{n}$, in this order, such that $A_{0} A_{1}=10 \text{~cm}, A_{1} A_{2}=2 \cdot A_{0} A_{1}, A_{2} A_{3}=3 \cdot A_{0} A_{1}, \ldots, A_{n-1} A_{n}=n \cdot A_{0} A_{1}$. Find the natural number $n$, such that $A_{0} A_{n}=2016 \cdot 10085$.

Monica Guita
3. $A_{0} A_{n}=A_{0} A_{1}+A_{1} A_{2}+A_{2} A_{3}+\ldots A_{n-1} A_{n}$ ..... (2p)
$A_{0} A_{1}+A_{1} A_{2}+A_{2} A_{3}+\ldots A_{n-1} A_{n}=A_{0} A_{1} \cdot(1+2+3+\ldots+n)=10 \cdot \frac{n(n+1)}{2}=5 n(n+1)$ ..... (3p)
$A_{0} A_{n}=2016 \cdot 2017 \cdot 5$ ..... (1p)
$n=2016$ ..... (1p)