2025 HCI P1 Q2

A sequence of real numbers, \({{x}_{n}}\), satisfies the recurrence relation

\({{x}_{n+1}}=a{{\left( -3 \right)}^{n}}+b{{n}^{2}}+c{{x}_{n}}\), for \(n\in \mathbb{Z}\), \(a\in \mathbb{Z}\), \(b\in \mathbb{Z}\), \(c\in \mathbb{Z}\) and \(n\ge 1\).

Given that \({{x}_{1}}=1\), \({{x}_{2}}=-8\), \({{x}_{3}}=70\), and \({{x}_{4}}=-377\), find the values of \(a\), \(b\) and \(c\).[3]

Hence find the values of \({{x}_{10}}\) and \({{x}_{11}}\).[2]