Solution
= Solution
For the symmetric increment $\xi_{n+1}$,
$$
\mathbb E(w^{X_{n+1}}\mid\mathcal F_n)
=w^{X_n}\frac{w+w^{-1}}2.
$$
The same multiplier holds for $w^{-X_n}$. Thus choose
$$
\boxed{z=\frac2{w+w^{-1}}>0}.
$$
Then each of $z^nw^{X_n}$ and $z^nw^{-X_n}$ is a martingale, and every linear combination
$$
(Aw^{X_n}+Bw^{-X_n})z^n
$$
is a martingale.