Suppose $X_1,\ldots,X_n$ are independent random variables with $E(X_i)=\mu$ and $\operatorname{Var}(X_i)=\sigma^2$ for all $i=1,\ldots,n$. Let $S_k=X_1+\cdots+X_k$. Find $\rho(S_k,S_n)$.
So I know that $\rho(S_k,S_n)=\operatorname{Cov}(S_k,S_n)/\sigma_k \sigma_n$ and $\operatorname{Cov}(S_k,S_n)=E(S_kS_n)-E(S_k)E(S_n)$. Since $S_k=X_1+\cdots+X_k$ I assume that $S_n=X_1,\ldots,X_n$. I'm not sure how to start this.