Cannot find recursive expression for $p_k$, when $p_{k+2}-p_k = λ(p_k-p_{k-1})$

Question

Given that $$ p_{k+2}-p_k = λ(p_k-p_{k-1}) \; , \; k \in \mathbb{N}\; , \; p_0 = 0 \; , \; p_{1000000} = 1 $$ can we find a recursive expression for $p_k$ ?

I found that $$ p_{2k+2}+p_{2k+1}-λp_{2k}=p_1+p_2 $$ How can I proceed from here?

Thank you in advance

Answer 1

Let $a_k=p_{k}-p_{k-1}$, then the given relation can be written as $$a_{k+2}+a_{k+1}=\lambda a_k.$$ Assuming $\lambda$ is a constant. You can now assume that $a_{k}=r^k$ and then the characteristic equation is $r^2+r-\lambda=0$. Hopefully you can go from here.

Answer 2

Hint: rewrite your recurrence as \begin{eqnarray*} p_{k+2} \color{blue}{- p_{k+1} +p_{k+1} } -p_k = \lambda (p_k-p_{k-1}) \end{eqnarray*} and let $q_{k+1}=p_{k+1}-p_k$.