smc

The sequence $x_1,x_2,\dots ,x_n$ is an arithmetic sequence since every term is $\frac{1}{2}$ more than the previous term. Letting $a=1$ and $r=\frac{1}{2}$, we can rewrite the sequence as $a,a+r,\dots ,a+(n-1)r$. Recall that the sum of the first $n$ terms of an arithmetic sequence is $na+\left(\genfrac{}{}{0ex}{}{n}{2}\right)r$. Substituting our values for $a$ and $r$, we get $n+\frac{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}{2}$. Simplifying gives $\boxed{\frac{n^2+3n}{4}}$