Problem: Suppose that $X_1,X_2,\dots$ are i.i.d. random variables with $E[X_1]=1$ and $E[X_1^2]=5$. Let $S_n=X_1+\cdots+X_n$. Compute $\text{Cov}(S_a,S_b)$ for $1\leq a<b.$
Attempt: By the bilinearity of the covariance we have $$\text{Cov}(S_a,S_b)=\sum_{i=1}^a\sum_{j=1}^b\text{Cov}(X_i,X_j).$$ Observe that if $i\ne j$ then since the random variables are i.i.d. we have $$\text{Cov}(X_i,X_j)=E[X_iX_j]-E[X_i]E[X_j]=E[X_i]E[X_j]-E[X_i]E[X_j]=0.$$ On the other hand if $i=j$ then $$\text{Cov}(X_i,X_i)=\text{Var}(X_i)=E[X_i^2]-E[X_i]^2=4.$$ Since $1\leq a<b$, it follows that $$\text{Cov}(S_a,S_b)=\sum_{i=1}^a\sum_{j=1}^b\text{Cov}(X_i,X_j)=4a.$$
Could anyone help me verify if the calculation above is correct?
Thank you for your help and your time and really appreciate any feedback.
As has already been stated in the comments, your approach and result are correct.
That independence implies absence of correlation (and thus zero covariance) is a standard fact that you probably don’t have to show with an explicit calculation from the expectation values (but of course that depends on the context).