How would I express the number with periodic decimal expansion 0.321212121... in the form $\displaystyle \frac{m}{n}$. $\displaystyle m,n \in \mathbb{Z}$ ?
I don't know how this is done.
Pretty much. The idea is to get rid of the "tail" depend on how many digits are repeating. e.g. In this case there are 2 digit repeating, so the factor will be 1000 and 10.
So say if the question is asking for a expression for 0.3214214214214...
Then, what do you think you should do?