2.5 Algebra Number Theory

Let a sequence be defined by \( a_0 = 1, a_1 = 1 \) and \( a_{n+2} = 3a_{n+1} + 4a_n \) for \( n \ge 0 \).
Prove by induction that \( a_n \equiv (-1)^n \pmod 5 \) for all \( n \ge 0 \).

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