Question 11
A sequence of real numbers: $a_0, a_1, a_2, \dots, a_{2012}$ satisfies the following conditions:
$$
|a_0 - a_1| = \frac{3^1}{2^0} |a_1 - a_2| = \frac{3^2}{2^1} |a_2 - a_3| = \dots = \frac{3^{2011}}{2^{2010}} |a_{2011} - a_{2012}| = \frac{3^{2012}}{2^{2011}} |a_{2012} - a_0|.
$$
Which values the subtraction: $a_0 - a_{1006}$ can take?
$$
|a_0 - a_1| = \frac{3^1}{2^0} |a_1 - a_2| = \frac{3^2}{2^1} |a_2 - a_3| = \dots = \frac{3^{2011}}{2^{2010}} |a_{2011} - a_{2012}| = \frac{3^{2012}}{2^{2011}} |a_{2012} - a_0|.
$$
Which values the subtraction: $a_0 - a_{1006}$ can take?