If $k = \frac{1}{1+2x}$, where $x$ is an integer greater than $1$ and $k$ can be represented as a terminating decimal, find the sum of all possible values of $k$.
I know that there are a lot of values for k, because there is an infinite amount of terminating decimals. However, I need to find the sum of them. I tried listing most of them, but I got nowhere. Can anyone guide to me to find the solution for this problem?