Have you ever made the limit of a sequence of matrices?

Geometry Level 3

If A=(5/2131)A = \begin{pmatrix} 5/2 & -1 \\ 3 & -1 \end{pmatrix}

what is limnAn\displaystyle \lim_{n \to \infty} A^n?

1=(1/2101), 2=(1234), 3=(4032)\boxed{1} = \begin{pmatrix} 1/2 & 1 \\ 0 & -1 \end{pmatrix}, \space \boxed{2} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \space \boxed{3} = \begin{pmatrix} 4 & 0 \\ 3 & 2 \end{pmatrix} 4=(4263), 5=(1240), 6=(2346)\boxed{4} = \begin{pmatrix} 4 & -2 \\ 6 & -3 \end{pmatrix}, \space \boxed{5} = \begin{pmatrix} 1 & 2 \\ 4 & 0 \end{pmatrix}, \space \boxed{6} = \begin{pmatrix} 2 & 3 \\ 4 & 6 \end{pmatrix} 7=(1/21/201), 8=(11/232), 9=(1003)\boxed{7} = \begin{pmatrix} 1/2 & 1/2 \\ 0 & 1 \end{pmatrix}, \space \boxed{8} = \begin{pmatrix} 1 & 1/2 \\ 3 & 2 \end{pmatrix}, \space \boxed{9} = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}

Example.- If you want to answer 5\boxed{5}, return 55.

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