Conditions for a linear estimator of the slope to be unbiased

Consider the simple linear regression model with $n$ random observations $Y_i=\beta_0+\beta_1x_i+\varepsilon_i$, $i=1,…,n$, $(n>2)$. $\beta_0$ and $\beta_1$ are unknown parameters, $x_1,…,x_n$ are observed values of the regressor variable and $\varepsilon_1,…,\varepsilon_n$ are error random variables with $E(\varepsilon_i)=0$, $i=1,…,n$, and for $i,j=1,…,n$, $Cov(\varepsilon_i,\varepsilon_j)=\begin{cases}0, & i\ne j,\ \sigma^2, & i=j.\end{cases}$ For real constants $a_1,…,a_n$, if $\sum_{i=1}^na_iY_i$ is an unbiased estimator of $\beta_1$, then