Clear[A]

A[n_] = {{1, 1/n}, {-1/n, 1}}

A[2]

A[2] // MatrixForm

A[2][[2, 1]]

Clear[B]

B[n_, k_] = MatrixPower[A[n], k]

B[4, 2] // MatrixForm

B[4, 2] . {0, 1} // MatrixForm

Clear[p]

p[n_, k_] = B[n, k] . {0, 1}

Clear[x]

x[n_, k_] = k/n

Clear[y]

y[n_, k_] = p[n, k][[1]]

N[x[5, 3]]

N[y[5, 2]]

Clear[T]

T[n_, a_] := Table[{N[x[n, m]], N[y[n, m]]}, {m, 0, a n}]

T[5, 1]

Clear[G]

G[n_, a_] := 
      ListPlot[T[n, a], 
        PlotJoined -> True, 
PlotStyle -> 
          Hue[0.6], AspectRatio -> Automatic, 
        PlotRange -> {-3, 2}]

G[30, 6]

Clear[G1]

G1[a_] := 
      Plot[Sin[x], {x, 0, a}, PlotStyle -> Hue[0.0], 
        AspectRatio -> Automatic, 
        PlotRange -> {-3, 2}]

G1[6]

Show[G[10, 6], G1[6]]

Do[Show[G1[6], G[n, 6]], {n, 2, 44, 6}]