[Math] How to find the limit of the sequence $x_n =\frac{1}{2}[x_{n-1}+x_{n-2}]$, if $x_0=0$ and $x_1=1$

Question

The formula is only applicable on values for $n\geq 2$.

I know that the sequence is monotonic with a lower bound at $\frac 1 2$, but I am unsure how to find the supremum of the sequence.

EDIT: $x_2 = \frac 1 2, x_3 = \frac 3 4, x_4 = \frac 5 8$. Does that mean that this sequence is only recursive and not monotonic?

Answer 1

You have: $$\begin{bmatrix}x_n\\x_{n-1}\end{bmatrix}=\begin{bmatrix}1/2&1/2\\1&0\end{bmatrix}\begin{bmatrix}x_{n-1}\\x_{n-2}\end{bmatrix}$$

And given the initial condition, $$\begin{align} \begin{bmatrix}x_n\\x_{n-1}\end{bmatrix} &=\begin{bmatrix}1/2&1/2\\1&0\end{bmatrix}^{n-1}\begin{bmatrix}1\\0\end{bmatrix}\\ \end{align}$$

Diagonalize (or rather, convert to Jordan Normal form) the matrix and you can give an explicit formula for $x_n$, from which the limit is clear.