Consider the sequence defined recursively by $a_1=-1$, $a_2=1$, $a_{n+1}=-15 a_{n-1}+8 a_n$. We can use matrix diagonalization to find an explicit formula for $a_n$.
a. Find a matrix that satisfies $\left[ \begin{array}{c} a_n \\ a_{n+1} \end{array} \right] = M \left[ \begin{array}{c} a_{n-1} \\ a_n \end{array} \right]$
$M=$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
b. Find the appropriate exponent $k$ such that
$\left[ \begin{array}{c} a_n \\ a_{n+1} \end{array} \right] = M ^ { k } \left[ \begin{array}{c} a_1 \\ a_2 \end{array} \right]$
$k=$
c. Find a diagonal matrix $D$ and an invertible matrix $P$ such that $M = P D P^{-1}$.
$D =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$, $P =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$,
d. Find $P^{-1}$.
$P^{-1} =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
e. Find $M^{5}=PD^{5}P^{-1}$.
$M^{5} =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$
f. Use parts b. and e. to find $a_{6}$.
$a_{6} =$
g. Develop an explicit formula for $a_n$ using part b. and a formula for $M^k = PD^kP^{-1}$.