A Neat Random Walk Problem

As many of the people that read this blog probably know, I enjoy inventing math problems, especially problems that are probabilistic in nature, and then working out the math to solve that problem. In this blog post, I will introduce one problem that I solved that I think is particularly elegant:
You are randomly walking along the x-axis starting at $x=0$. When you are at $x = n$, you move to $x = n-1$ with probability $p = \frac{n}{n+1}$ and to $x = n+1$ with probability $p = \frac{1}{n+1}$. What is the expected number of steps until you return to $x = 0$ for the first time?
This problem differs from standard random walk problems because the probability of moving left and right along the x-axis depends on your current location. In this problem, there is a force that is pulling you towards $x = 0$, and the further away you get, the more of an effect this force has on you.