<s>[INST] 2 Related Questions About Finding A Closed Form 
*

*Consider the sequence defined by
$$
\begin{cases}
s_0=0\\
s_1=3\\
s_n=6s_{n-1}-9s_{n-2} & \text{if }n\ge 2
\end{cases}
.$$
Find a closed form for $s_n$.





*Consider the sequence defined by
$$
\begin{cases}
t_0=5\\
t_1=9\\
t_n=6t_{n-1}-9t_{n-2} & \text{if }n\ge 2
\end{cases}
.$$
Find a closed form for $t_n$.



I am having trouble with these questions about closed forms, could someone walk me step by step through each problem?  Thanks!
 [/INST] Hint: 
For you first problem you could rewrite it as a matrix equation
\begin{align}
\begin{pmatrix}
s_n\\
s_{n-1}
\end{pmatrix}
=
\begin{pmatrix}
6s_{n-1}-9s_{n-2}\\
s_{n-1}
\end{pmatrix}
=
\begin{pmatrix}
6 &-9\\
1 & 0
\end{pmatrix}
\begin{pmatrix}
s_{n-1}\\
s_{n-2}
\end{pmatrix}
=
\begin{pmatrix}
6 &-9\\
1 & 0
\end{pmatrix}^{n-1}
\begin{pmatrix}
s_{1}\\
s_{0}
\end{pmatrix}.
\end{align}
Thus, to find the closed form of $s_n$ it suffices to compute the $n-1$ power of your matrix. 
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