Showing that an unbiased estimator exists for a distribution with given μ

Question

While studying mathematical statistics I stumbled upon this exercise in a book. Since I'm just beginning to learn statistics I don't fully understand the problem. Sadly the book doesn't provide any further informations that could help me come to grips it.

(my translation )

"Trait being examined has a distribution with given μ as a expected value. Show that if $X_1$,..,$X_n$ is a sample, then the statistics of a form $\mu^{*}$ = $ \sum\limits_{k=1}^\mathbb{n}a_nX_n$ , where $a_1$+...+$a_n$=1 is unbiased estimator of μ. What is the interpretation of this fact?"

Answer 1

An estimator $\mu^\ast$ of $\mu$ is called unbiased if $\mathbb{E} [\mu^\ast] = \mu$, which means that the estimator is correct "on average".

So you would have to show that $ \mathbb{E}\left[\sum_{k=1}^n a_n X_n\right] = \mu. $