Question 2
Let $P(x)=rx^3+qx^2+px+1$, where $p,q,r$ are real and $r\gt 0$. Define $(a_n)$ by
$$a_1=1,\qquad a_2=-p,\qquad a_3=p^2-q,$$
$$a_{n+3}=-pa_{n+2}-qa_{n+1}-ra_n\qquad(n\ge0).$$
Prove that if $P$ has exactly one real root and has no repeated root, then $(a_n)$ has infinitely many negative terms.
Editorial clarification: the source prints $n\ge0$ although $a_0$ is not listed. Its equation at $n=0$ forces $a_0=0$ because $r\gt 0$. Equivalently, use the three displayed initial terms and apply the recurrence for $n\ge1$.
$$a_1=1,\qquad a_2=-p,\qquad a_3=p^2-q,$$
$$a_{n+3}=-pa_{n+2}-qa_{n+1}-ra_n\qquad(n\ge0).$$
Prove that if $P$ has exactly one real root and has no repeated root, then $(a_n)$ has infinitely many negative terms.
Editorial clarification: the source prints $n\ge0$ although $a_0$ is not listed. Its equation at $n=0$ forces $a_0=0$ because $r\gt 0$. Equivalently, use the three displayed initial terms and apply the recurrence for $n\ge1$.