Given a gambler's ruin problem where the outcome of each bet is a loss of $1$ with $q = 0.7$ and a win of $2$ with $p = 0.3$. Find the explicit formula for reaching state $10$, starting with $n$ dollars ($n = 0,1,2,...,10$).
My attempt: Let $P_i =$ probability of reaching state $10$ starting in state $i$. Then condition on the first step starting in state $i$, we have:
$P_{i} = p P_{i+2} + q P_{i-1}$ for $i=1,2,\ldots, 9$ ($P_{11} = P_{10}$ since from state $i$, reaching state $11$ is also the same as reaching state $10$).
After I tried very hard to manipulate this recursion, I ended up with $2p_{10} - p_1 - p_2 = \frac{q}{p}P_9$. Using this together with the fact that $\sum_{i=1}^{10}P_i = 1$, we have 2 equations but 3 unknown variables $p_1, p_2, p_9$, so I'm stumbled here.
My question: How could we solve for $p_1$ and $p_2$? Did I set up the recursion incorrectly?