A bit of a contrived example, but if I had a sample of $X_1,\dots,X_n \sim N(\mu,\sigma^2)$ (in this case $\mu$ is unknown but $\sigma^2$ is known), and then calculated the arithmetic mean of the sample, how do I find the covariance between $$Cov(X_1,\overline{X})$$

I know (at least I think I do) that it should become equal to Var$(\overline{X})$, which I can state as $\frac{\sigma^2}{n}$ because $\sigma^2$ is known.

I rewrite as:

$$Cov(X_1,\overline{X})=E(X_1\overline{X})-E(X_1)E(\overline{X})$$

which becomes $E(X_1\overline{X})-E(\overline{X})^2$ but evaluating that first part is causing me some issues. Am I on the right track?

Thank you.