35. Consider integers $\{1,2, \ldots, 10\}$. A particle is initially at 1. It moves to an adjacent integer in the next step. What is the expected number of steps it will take to reach 10 for the first time?
35. Answer: 81 .
Let $E_{k}$ denote the expected number of steps it takes to go from $k-1$ to $k, k=2, \ldots, 100$. Then $E_{k+1}=\frac{1}{2}\left(1+E_{k}+E_{k+1}\right)+\frac{1}{2}$, which implies $E_{k+1}=E_{k}+2$.
It is clear that $E_{2}=1$. Then $E_{3}=3, E_{4}=5, \ldots, E_{10}=17$. So
$$
E=E_{2}+E_{3}+\cdots+E_{10}=1+3+\cdots+17=81 .
$$