Let us consider the sample of independent random variables $X_1,\ldots,X_n$ with $E[X_1]=E[X_2]\ldots=E[X_n]=\mu$. We examine the following estimator of $\mu$: $$a_1(X_1+X_2+\ldots+X_n)+a_2.$$ Find constants $a_1$ and $a_2$ such that the estimator is unbiased.
I proceed as follows:
$$E[a_1(X_1+X_2+\ldots+X_n)+a_2]=a_1E[X_1+X_2+\ldots+X_n]+a_2=a_1n\mu+a_2.$$
So estimator is unbiased for $a_1=1/n$ and $a_2=0$ since the above expected value is equal to $\mu$.
I wonder if I should separately consider the case when $a_2\neq 0$ and find $a_1$ such that the expected value is equal to $\mu$.