2.5 Algebra Number Theory

Consider the sequence defined by \( x_1=1, x_2=3 \) and \( x_n = 4x_{n-1} - x_{n-2} \) for \( n \ge 3 \).
Prove by induction that \( x_n^2 - 4x_n x_{n-1} + x_{n-1}^2 = -3 \) for all \( n \ge 2 \).

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