Markov Chain discarding balls from urn

Question

The following question has me stumped. Any ideas on how to get started?

An urn contains $n$ green balls and $n+2$ red balls. A ball is picked at random: if it is green then a red balls is also removed and both are discarded; if it is red, then it is replaced together with an extra red and an extra green ball. This is repeated until there are no green balls in the urn. Show that the probability the process terminates is $1/(n+1)$.

Answer 1

Hint: Let $P_n$ be the probability that with $n$ green balls and $n+2$ red balls, the process terminates.

Hint: Write the markov process for $n\geq 1$. Show that

$$ P_n = \frac{n}{2n+2} P_{n-1} + \frac{n+2}{2n+2} P_{n+1} $$

Hint: Show that $P_0 = 1$. Use the above relation to conclude that $P_n = \frac{1}{n+1}$ by induction.