$[8]$ Consider the sequence $y_n$ given recursively by $y_1=1,y_2=3,$ and

$y_n=\frac{1}{3}{y}_{n-1}+\frac{2}{3}{y}_{n-2}$

for $n>2.$

$($a$)[2]$ Show that $y_n$ is a contractive sequence. Thus, $y_n$ is convergent.

$($b$)[2]$ Express $y_n-y_{n-1}$ in terms of $y_2-y_1.$

$($c$)[2]$ Use $($b$)$ to express $y_n-y_1$ in terms of $y_2-y_1.$

$($d$)[2]$ Use $($c$)$ to find $\underset{?}{lim}{y}_n.$