npoints = 100 ;
r = 3.5 ;
f[x_] := r x (1-x) ;

series = Table[ 0., {npoints} ] ;

series[[1]] = 0.2 ;
For[ i = 1, i < npoints, i++,
series[[i+1]] = f[ series[[i]] ] ;
] ;

ListLinePlot[ series, PlotRange -> {0.,1.} ]