Clear[A,  P,  Initial]
A  =  {{0.30,  0.80},  {0.90,  0.10}}
Initial  =  {{1},  {1}}
P[n_]  :=  
      If[n  ==  1,  Initial,  A  .  P[n  -  1]]

P[10][[1, 1]]

P[20]

ListPlot[
Table[{P[n][[1,  1]],  P[
              n][[2,  1]]},  {n, 1,  10}],  
      PlotRange  ->  {{0,  5000},  {0,  
            5000}}, 
PlotStyle  ->  
        PointSize[0.015]]

TableForm[
      Table[{P[n][[1,  1]],  P[n][[2,  1]]},  {n, 1,  
          10}]]

Eigensystem[A]

Clear[Q]
Q[n_]  :=  P[n][[1,  1]]  +  P[n][[2,  
            1]]
TableForm[
      Table[{P[n][[1,  1]]/Q[n],  P[n][[2,  1]]/
            Q[n]}, 
 {n, 1,  20}]]

TableForm[Table[Q[n + 1]/Q[n],  {n,  1,  20}]]