Question 129
Consider $a, b \in \mathbb{N}^*$. Define the sequence $(x_n)_{n \in \mathbb{N}}$ by $x_0 = 0, x_1 = 1$ and
$$
x_{n+2} = a x_{n+1} + b x_n, \quad \forall n \in \mathbb{N}
$$
and the matrix $A$, by
$$
A = \begin{bmatrix} a & b \\ 1 & 0 \end{bmatrix}.
$$
a) Prove that
$$
A^n = \begin{bmatrix} x_{n+1} & b x_n \\ x_n & b x_{n-1} \end{bmatrix}, \quad \forall n \in \mathbb{N}^*.
$$
$$
x_{n+2} = a x_{n+1} + b x_n, \quad \forall n \in \mathbb{N}
$$
and the matrix $A$, by
$$
A = \begin{bmatrix} a & b \\ 1 & 0 \end{bmatrix}.
$$
a) Prove that
$$
A^n = \begin{bmatrix} x_{n+1} & b x_n \\ x_n & b x_{n-1} \end{bmatrix}, \quad \forall n \in \mathbb{N}^*.
$$