Identify the $xy$-plane with the complex plane $\mathbb{C}$, so that $P_k$ is the real number $k$. If $z$ is sent to $z'$ by a counterclockwise rotation by $\theta$ about $P_k$, then $z'-k = e^{i\theta} (z-k)$; hence the rotation $R_k$ sends $z$ to $\zeta z + k (1-\zeta)$, where $\zeta = e^{2\pi i/n}$. It follows that $R_1$ followed by $R_2$ sends $z$ to $\zeta(\zeta z +(1-\zeta)) + 2 (1-\zeta) = \zeta^2 z + (1-\zeta)(\zeta + 2)$, and so forth; an easy induction shows that $R$ sends $z$ to $$\zeta^n z + (1-\zeta)(\zeta^{n-1} + 2 \zeta^{n-2} + \dots + (n-1) \zeta + n).$$ Expanding the product $(1-\zeta)(\zeta^{n-1} + 2 \zeta^{n-2} + \dots + (n-1) \zeta + n)$ yields $-\zeta^n - \zeta^{n-1} - \dots - \zeta + n = n$. Thus $R$ sends $z$ to $z+n$; in cartesian coordinates, $R(x,y) = (x+n,y)$.