Solve $\frac{1}{2x}+\frac{1}{2}\left(\frac{1}{2x}+\cdots\right)$ If
$$\displaystyle \frac{1}{2x}+\frac{1}{2}\left(\frac{1}{2x}+ \frac{1}{2}\left(\frac{1}{2x} +\cdots\right) \right) = y$$
then what is $x$?
I was thinking of expanding the brackets and trying to notice a pattern but as it effectively goes to infinity. I don't think I can expand it properly, can I?

Expand. The first term is $\frac{1}{2x}$.
The sum of the first two terms is $\frac{1}{2x}+\frac{1}{4x}$.
The sum of the first three terms is $\frac{1}{2x}+\frac{1}{4x}+\frac{1}{8x}$.
And so on.
The sum of the first $n$ terms is 
$$\frac{1}{2x}\left(1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^{n-1}}\right).$$
As $n\to\infty$, the inner sum approaches $2$. So the whole thing approaches $\frac{1}{x}$.
So we are solving the equation $\frac{1}{x}=y$.
