Solution
= Solution
If $\xi_k=S_k-S_{k-1}$, then
$$
\mu=\mathbb E\xi_k=p-(1-p)=2p-1.
$$
The centered increments $\xi_k-\mu$ are independent of the past and have mean zero, so
$$
\boxed{M_n=S_n-\mu n}
$$
is a <martingale>.
= Solution
If $\xi_k=S_k-S_{k-1}$, then
$$
\mu=\mathbb E\xi_k=p-(1-p)=2p-1.
$$
The centered increments $\xi_k-\mu$ are independent of the past and have mean zero, so
$$
\boxed{M_n=S_n-\mu n}
$$
is a <martingale>.