p[k_,n_]:=((k-2) n (n+1))/2-(k-3) n;
Manipulate[nums=Table[p[k,n],{n,1,numpoints}];
If[showpoints,ListPolarPlot[Table[{2*Pi*Sqrt[n],Sqrt[n]},{n,nums}],Joined->joinpoints,Axes->False,PlotMarkers->{Automatic,Small},ImageSize->{400,400}],ListPolarPlot[Table[{2*Pi*Sqrt[n],Sqrt[n]},{n,nums}],Joined->joinpoints,Axes->False,ImageSize->{400,400}]],{k,2,50,Appearance->"Labeled"},{{numpoints,100,"Number of points"},{100,250,500}},{{showpoints,True,"Show points"},{True,False}},{{joinpoints,True,"Join points"},{True,False}},SaveDefinitions:>True]