moment generating function for $S_N=X_1+\cdots+X_N$ with $N$ dependent of $X_1$

Question

Let $X_1, X_2, \ldots$ independent and identically distributed discrete random variables with support contained in $\mathbb{N}$ and let $N=X_1+1$. How can I calculate the moment generating function of $$S_N=X_1+\cdots+X_N?$$

Thanks for any help.

My attepmt: We have that $$M_{S_N}(t)=E[e^{rS_N}]=E[e^{r0}]Pr(N=0)+\sum_{n=1}^\infty E[e^{r(X_1+\cdots+X_n)}|N=n]Pr(N=n) $$

Answer 1

$Ee^{tS_N}=\sum_n EI_{X_1=n}e^{t(X_1+X_2+..+X_{n+1})}=\sum_n M(t)^{n}P(X_1=n)e^{tn}=\sum_n P(X_1=n)e^{(t+\log(M(t))n}=M(t+\log(M(t))$.