Adapted from SCSU Math Contest

$\displaystyle F(n + 1) = \frac{2F(n) + 1}{2} = F(n) + \frac 12$

This is simply an arithmetic sequence with $d = \frac 12$ and $a_1 = 2$ .

$\begin{aligned} \displaystyle a_{2017} & = 2 + \frac 12(2016) & \small \color{#3D99F6} a_n = a_1 + d(n - 1) \\ & = 1010 \end{aligned}$