(Subscript[x, 0]) = 2.;

For[k=0, k<20, (Subscript[x, k+1]) = (1/2)((Subscript[x, k])+ (5/(Subscript[x, k])));Print[(Subscript[x, k])],k++; ]

N[Sqrt[5]]

PtsToPlot = Table[{n,(Subscript[x, n])},{n,0,19}];

Plot1 = ListPlot[PtsToPlot,PlotMarkers->{"x",Medium},Axes-> True, AxesOrigin->{0,2}, PlotRange->{{0,20},{1.9,2.5}}];

LinePlot =Graphics[{Thickness[.001],Red,Line[{{0,Sqrt[5]},{20,Sqrt[5]}}]}];

Show[Plot1,LinePlot]

PtsDiff = Table[{n,Abs[(Subscript[x, n])-Sqrt[5]]},{n,0,19}];

ListPlot1 = ListPlot[PtsDiff ,PlotMarkers->{Automatic,Medium},Axes-> True, AxesOrigin->{-0.5,0}, PlotRange->{{-0.55,19},{-0.1,0.3}}]

(Subscript[y, 0]) = 5.0;

For[k=0, k<20, (Subscript[y, k+1]) = (1/2)((Subscript[y, k])+ (5/(Subscript[y, k])));Print[(Subscript[y, k])],k++; ]

PtsToPloty = Table[{n,(Subscript[y, n])},{n,0,19}];

Plot2 = ListPlot[PtsToPloty,PlotMarkers->{"y",Medium},Axes-> True, AxesOrigin->{0,2}, PlotRange->{{0,20},{1.9,5.2}}];

Show[Plot1,Plot2, PlotRange->{{0,20},{1.9,5.2}}]

PtsDiffy = Table[{n,Abs[(Subscript[y, n])-Sqrt[5]]},{n,0,19}];

ListPlot[{PtsDiff,PtsDiffy} ,PlotMarkers->{Automatic,Medium},Axes-> True, AxesOrigin->{-0.5,0}, PlotRange->{{-0.55,19},{-0.1,0.3}}]

(Subscript[z, 0]) = -5.0;

For[k=0, k<20, (Subscript[z, k+1]) = (1/2)((Subscript[z, k])+ (5/(Subscript[z, k])));Print[(Subscript[z, k])],k++; ]

PtsToPloty = Table[{n,(Subscript[z, n])},{n,0,19}];

Plot3 = ListPlot[PtsToPloty,PlotMarkers->{"z",Medium},Axes-> True, AxesOrigin->{0,0}, PlotRange->{{0,20},{1.9,5.2}}];

Show[Plot1,Plot2, Plot3, PlotRange->{{0,20},{-5.2,5.2}},AxesOrigin->{0,0}]

Plot[{(1/2) (x+(5/x)),x},{x,-10,10}]

Plot[{(1/2) (x+(5/x)),x},{x,0,10}, PlotRange->{{0,10},{0,10}}, AspectRatio->1]

Clear[x]

Solve[{x==(1/2)(x+5/x)},x]