A={{1,1/2,1/4},{1/4,0,1},{1,-1/2,0}};
B={1,1/10,1/2};
x0={25,0,-25};

C1=Transpose@Table[MatrixPower[A,19-i].B,{i,0,19}];
F1=LowerTriangularize@ConstantArray[1,{20,20}];

solution=LeastSquares[C1.F1,-MatrixPower[A,20].x0];
F1.solution//MatrixForm//N