In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Give the general form of sequences belonging to $\operatorname { Sol } ($ II.1 $)$ as a function of the complex roots $r _ { 1 }$ and $r _ { 2 }$ of the equation $r ^ { 2 } + a r + 1 = 0$. What are $r _ { 1 } + r _ { 2 }$ and $r _ { 1 } r _ { 2 }$?
In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation
$$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$
Give the general form of sequences belonging to $\operatorname { Sol } ($ II.1 $)$ as a function of the complex roots $r _ { 1 }$ and $r _ { 2 }$ of the equation $r ^ { 2 } + a r + 1 = 0$. What are $r _ { 1 } + r _ { 2 }$ and $r _ { 1 } r _ { 2 }$?