Binary expansion 
The ternary expansion $x = 0.10101010\ldots$ is given. Give the binary expansion of $x$. Alternatively, transform the binary expansion $y = 0.110110110\ldots$ into a ternary expansion.

By the ternary expansion, do they mean $0.10101010\ldots_3$? Otherwise it just seems like we need to find the binary expansion of $0.10101010\ldots.$
 A: Hints:
In decimal if a number repeats with periodicity one after the decimal point, it can be achieved via division by nine by the repeated number.
E.g. $\frac{2}{9}=0.222222222\dots$
$\frac{5}{9}=0.555555555\dots$
If it has periodicity two, it can be accomplished via division by $99$
$\frac{13}{99}=0.1313131313\dots$
$\frac{57}{99}=0.5757575757\dots$
If it has periodicity longer than that, use more nines...

What is special about nine in this pattern?

 It is one less than ten.

How might this pattern extend to other number bases?

 Dividing by the number one less than the base has a similar pattern.  E.g. $\frac{3}{4}=0.3333333\dots$ in base five.  Similarly $\frac{42}{77}=0.42424242\dots$ in base eight (octal).  (That is specifically $42_8$ divided by $77_8$, i.e. $\frac{34}{63}$ in decimal.)


For your number, $0.10101010\dots_3$ what might it be if expressed as a fraction?

 $\frac{10}{22}$ with both $10$ and $22$ in base $3$., i.e. $\frac{1\cdot 3+0}{2\cdot 3+2} = \frac{3}{8}$ in decimal.

