Find the value of expression!

Let the sum be $S$ . Then we have:

$\begin{aligned} S & = \sum_{n=1}^{34} \left \lfloor \frac {18n}{35} \right \rfloor \\ & = \sum_{n=1}^{34} \left \lfloor \frac {(17.5+0.5)n}{35} \right \rfloor \\ & = \sum_{n=1}^{34} \left \lfloor \frac n2 + \blue{\frac n{70}} \right \rfloor & \small \blue{\text{for }n < 70 \left \lfloor \frac n{70} \right \rfloor = 0} \\ & = \sum_{n=1}^{34} \left \lfloor \frac n2 \right \rfloor \\ & = 0 + 1 + 1 + 2 + 2 + \cdots + 16 + 16 + 17 \\ & = 2 \sum_{n=1}^{16} n + 17 \\ & = 2 \times \frac {16\times 17}2 + 17 \\ & = \boxed{289} \end{aligned}$