Suppose that $x_1, x_2,...x_n$ are independent copies of random variable $x$ having distribution $$P(x=1)=P(x=-1)=\frac{1}{2}.$$ In addition, suppose that $\mathcal{D}=\mathcal{D}_{x_1,...,x_k}(k=1,...,n),$ $S_k=x_1+x_2+...+x_k$ for $(k=1,...,n)$ and $\tau$ is a stopping time with respect to the decomposition sequence $\mathcal{D}_1 \preceq \mathcal{D}_2 \preceq ... \preceq \mathcal{D}_n.$ Show that $\mathbb{E}S_{\tau}^2=\mathbb{E}\tau$.
How should I start and what to use. I have no idea... I know that $\mathbb{E}S_k=\mathbb{E}x_1 \mathbb{E}k$. Do I need to show that $S_k$ is a martingale?