Problem

Source: 2019 Pan-African Mathematics Olympiad, Problem 1

Tags: recurrence relation, Linear Recurrences, algebra, induction, PAMO



Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows: $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$. Show that $a_n$ is always a strictly positive integer.