Problem: Let $X_1, ... ,X_n$ be a sequence of i.i.d. random variables which are $\mathrm{Pois}(\lambda).$ Let $\mathscr{F}_n=\sigma(X_1,...,X_n), n\geq 1$ and define
$$S_n =\displaystyle\sum_{i=1}^n X_i, \: \: M_n =S_n-n\lambda, \: \: \tau=\inf\{n\geq 1 : S_n \geq 2\}.$$
I have proved that Sn for $n\geq 0$ is a Markov Chain on the set of natural numbers with 0, and that Mn for $n\geq 1$ is a martingale with respect to the filtration $\mathscr{F}_n$ for $n\geq 1.$
I now have to show that $P(\tau >n)=e^{-n\lambda}(1+n\lambda)$.
But i'm having a hard time starting on this problem. I know that $S_n \sim\mathrm{ Pois}(n\lambda)$.
Any hints?