The range of a $U(\theta,\theta+1)$ sample has mean $(n-1)/(n+1)$

Let $X_1,X_2,\dots,X_n$ be iid r.v.s with $U(\theta,\theta+1)$, then $E[X_{(n)}-X_{(1)}]$, where $X_{(n)}=\max X_i$ and $X_{(1)}=\min X_i$, is given by: