AHSME 1962 Problem 32

If $x_{k+1} = x_k + \frac12$ for $k=1, 2, \dots, n-1$ and $x_1=1$, find $x_1 + x_2 + \dots + x_n$. $\textbf{(A)}\ \frac{n+1}{2} \textbf{(B)}\ \frac{n+3}{2} \textbf{(C)}\ \frac{n^2-1}{2} \textbf{(D)}\ \frac{n^2+n}{4} \textbf{(E)}\ \frac{n^2+3n}{4}$