How to determine if the process is martingale?

Question

I have that $\{X_t; t=1,2... \}$ is an i.i.d random variables such that $P(X_t=1)=p=1/4$ and $P(X_t=-1)=1-p=3/4$. Let $S_0=0$ and $S_t=\sum_{i=1}^tX_i$ for $t=1,2...$
Let finally $Y_t=a^{S_t}$ for $t=0,1,2...$ where $a$ is a constant (not zero).
The question is, for which values of $a$ is $Y_t; t=0,1,2..$ a martingale with respect to $X_t; t=0,1,2... ? \quad$
Could somebody help me with this? I have tried with the definition of a martingale $E[a^{\sum_ {i=1}^{t+1}X_i}|X_t, X_{t-1}...]$ but I am stuck here

Answer 1

$E[a^{\sum_ {i=1}^{t+1}X_i}|X_t, X_{t-1}...]=a^{\sum_ {i=1}^{t+1}X_i}Ea^{X_t}$. So we get a martinagle if and only if $Ea^{X_t}=1$ for all $t$. Can you compute $Ea^{X_t}$ and solve $Ea^{X_t}=1$ for $a$?