<s>[INST] Find the Limit.? What is the limit of
$Y_{n} := \frac{1}{3}Y_{n-1} + \frac{2}{3}Y_{n-2} $
for $n > 2$ and $Y_{1}<Y_{2}$.
I have tried solving, but the limit tends to exist between $y_{1}$ and $y_{2}$ and hence I am not able to evaluate what would be the ratio between which they would lie?
 [/INST] Let me show you the general method for dealing with these linear recurrences. The idea is to convert the linear recurrence into a matrix recurrence: $v_{n} = Av_{n-1}$ so that $v_n = A^nv_0$.
Here, you have
$$ \begin{pmatrix} Y_n \\ Y_{n - 1} \end{pmatrix} = \begin{pmatrix} \frac13 & \frac23 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} Y_{n - 1} \\ Y_{n - 2} \end{pmatrix} $$
So by induction,
$$ \begin{pmatrix} Y_n \\ Y_{n - 1} \end{pmatrix} = \begin{pmatrix} \frac13 & \frac23 \\ 1 & 0 \end{pmatrix}^{n - 2} \begin{pmatrix} Y_{2} \\ Y_{1} \end{pmatrix}. $$
The matrix
$$ P = \begin{pmatrix} \frac13 & \frac23 \\ 1 & 0 \end{pmatrix}$$
is a regular, row-stochastic matrix. Thus by standard Markov chain theory,
$$ P^n \to \begin{pmatrix} \frac35 & \frac25 \\ \frac35 & \frac25 \end{pmatrix} $$
where $(3/5, 2/5)$ is the probability vector in the 1-eigenspace of $P^T$. So $Y_n \to \frac25Y_1 + \frac 35Y_2$.
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