I need some help proving this because i couldn't reach the form the exercise asks for
to simplify the exercise we take that all fractions are defined \ Let $(u_n)$ and $(v_n)$ be two sequences defined by their first terms $u_0$ , $v_0$ $\in \mathbb{R}$ and the following relation $\forall n \in \mathbb{N}$ : $$ \begin{cases} u_{n+1}=\alpha u_n + \beta v_n \\ v_{n+1} = \lambda u_n + \mu v_n \end{cases} $$ let $(a_n)$ and be a sequence defined $\forall n \in \mathbb{N}$ as follows : $$ a_n = p_1 u_n + q_1 v_n $$
prove the following equivalence \begin{cases} p_1 = \dfrac{\mu - \lambda}{\alpha \mu - \beta \lambda} \\ q_1 = \dfrac{\alpha - \beta}{\alpha \mu - \beta \lambda} \end{cases} equivalent to $(a_n)$ being cosntant
all i've done is calculating $$a_{n+1}=(\alpha p_1 + \lambda q_1)u_n + (\beta p_1 + \mu q_1)v_n$$ and also having $$a_n = p_1 u_n + q_1 v_n$$ so for $(a_n)$ to be constant we must get $ a_{n+1} = a_n$ therefore : $$(\alpha p_1 + \lambda q_1)u_n + (\beta p_1 + \mu q_1)v_n = p_1 u_n + q_1 v_n$$ and from here i don't know what to go for