Answer
The average value is
$$\overline{f}=\frac{2}{\pi}.$$
Work Step by Step
The formula for the average value of the function $f$ on a segment $[x_1,x_2]$ is
$$\overline{f}=\frac{\int_{x_1}^{x_2}f(x)dx}{x_2-x_1}.$$
Using the values given in this problem we have $x_1=0$, $x_2=\pi/n$ and $f(x)=\sin nx$:
$$\overline{f}=\frac{\int_{0}^{\pi/n}\sin nxdx}{\pi/n-0}=\frac{n}{\pi}\int_{0}^{\pi/n}\sin nxdx=\frac{n}{\pi}\left.\left(-\frac{1}{n}\cos nx\right)\right|_0^{\pi/n}=\\-\frac{1}{\pi}\left(\cos n\frac{\pi}{n}-\cos 0\right)=-\frac{1}{\pi}(\cos\pi-\cos0) = \frac{2}{\pi}.$$
