Bias and consistency of a combination of a sample mean and one new observation

Let $X_1,X_2,…,X_n,X_{n+1}$ be a random sample from a $N(\mu,1)$ population. If $\overline X_n=\frac1n\sum_{i=1}^nX_i$ and $T=\frac12\left(\overline X_n+X_{n+1}\right)$, then for estimating $\mu$