, 2026

Let $N$ be a positive integer and let $(k_n)_{n \geq 1}$ be a sequence of positive integers with all terms at most $N$. Annabel begins by choosing integers $x_1, x_2, \ldots, x_N$. She then extends this to an infinite sequence $(x_n)_{n \geq 1}$ of integers by defining
$$x_n = \sum_{i=n-k_n}^{n-1} x_i$$
for each $n > N$.
Show that there are either finitely many strictly positive terms or finitely many strictly negative terms in the infinite sequence $(x_n)$.