Ex: Solve the recurrance relation: h_{n} + h_{n−2} = 0 h_{0} = 3, h_{1} = 5 q^{n} + q^{n−2} = 0 q^{2} + 1 = 0 q = ±i h_{n} = c_{1}i^{n} + c_{2}(−i)^{n} c_{1} + c_{2} = 3 implies c_{2} = 3 − c_{1} c_{1}i − c_{2}i = 5 −c_{1} + c_{2} = 5i −c_{1} + 3 − c_{1} = 5i −2c_{1} + 3 = 5i c_{1} = \frac{3−5i}{2} c_{2} = 3 − (\frac{3−5i}{2}) = \frac{3+5i}{2} h_{n} = (\frac{3−5i}{2})i^{n} + (\frac{3+5i}{2})(−i)^{n} h_{2j} = (\frac{3−5i}{2})i^{2j} + (\frac{3+5i}{2})(−i)^{2j} h_{2j+1} = (\frac{3−5i}{2})i^{2j+1} + (\frac{3+5i}{2})(−i)^{2j+1} = (\frac{3i+5}{2})(−1)^{j} + (\frac{−3i+5}{2})(−1)^{j} Ex: Solve the recurrance relation, h_{n} − 2h_{n−1} + h_{n−3} − h_{n−4} = 0 h_{0} = 3, h_{1} = 3, h_{2} = 7, h_{3} = 15, (x − 1)^{3}(x + 1) = (x^{3} − 3x^{2} + 3x − 1) = (x^{4} − 2x^{3} + 2x − 1) q = 1 , 1, 1, -1 h_{n} = c_{1} + c_{2}n + c_{3}n^{2} + c_{4}(−1)^{n} h_{0} = c_{1} + c_{4} h_{1} = c_{1} + c_{2} + c_{3} − c_{4} h_{2} = c_{1} + 2c_{2} + 4c_{3} + c_{4} h_{3} = c_{1} + 3c_{2} + 9c_{3} − c_{4}

\begin{matrix}1 & 0 & 0 & 1 & a \\ 1 & 1 & 1 & −1 & b \\ 1 & 2 & 4 & 1 & c \\ 1 & 3 & 9 & −1 & d \end{matrix}

\begin{matrix}1 & 0 & 0 & 1 & a \\ 0 & 1 & 1 & −2 & b − a \\ 0 & 2 & 4 & 0 & c − a \\ 0 & 3 & 9 & −2 & d − a \end{matrix}

\begin{matrix}1 & 0 & 0 & 1 & a \\ 0 & 1 & 1 & −2 & b − a \\ 0 & 0 & 2 & 4 & c − a − 2(b − a) \\ 0 & 0 & 6 & 4 & d − a − 3(b − a) \end{matrix}

\begin{matrix}1 & 0 & 0 & 1 & a \\ 0 & 1 & 1 & −2 & b − a \\ 0 & 0 & 2 & 4 & c − a − 2(b − a) \\ 0 & 0 & 0 & −8 & d − a − 3(b − a) − 3[c − a − 2(b − a)] \end{matrix}

1 0 0 1  a
0 1 1 −2
0 0 2 4  c − a
0 0 0 −8 d − a − 3[c − a]

\begin{matrix}1 & 0 & 0 & 1 & 3 \\ 0 & 1 & 1 & −2 & 0 \\ 0 & 0 & 2 & 4 & 4 \\ 0 & 0 & 0 & −8 & d − 3 − 12 \end{matrix}

h_{0} = 3, h_{1} = 3, h_{2} = 7, h_{3} = 15,

\begin{matrix}1 & 0 & 0 & 1 & 3 \\ 0 & 1 & 1 & −2 & 0 \\ 0 & 0 & 1 & 2 & 2 \\ 0 & 0 & 0 & 1 & 0 \end{matrix}

1 0 0 0 3
0 1 1 0 0
0 0 1 0 2
0 0 0 1 0

\begin{matrix}1 & 0 & 0 & 0 & 3 \\ 0 & 1 & 0 & 0 & −2 \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 1 & 0 \end{matrix}

= (x^{3} − 3x^{2} + 3x − 1)

= (x^{4} − 3x^{3} + 3x^{2} − x)