Expected Value/Probability

Question

The question is attached, I was solving problems on probability and came across this PS: it's not a homework problem

I was trying to first to generate a recursion as:

$$P(n)=P(n-1)/2 + P(n+1)/2,$$ where $P(n)$ is the probability to reach nth position. Now, calculating $P(1)$ , was getting tough since it's involving a lot of case work like reaching 1st position after 1st pass,3rd pass.. and so on

Please help.