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?

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 $Y_n \to \frac25Y_1 + \frac 35Y_2$.
