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If \( \alpha, \beta \) be the real roots of \( a x^{2}+b x+c=0 \) and \( s_{n}=\alpha^{n}+\beta^{n} \), then prove that \( a s_{n}+b s_{n-1}+c_{n-2}=0 \) for all \( n \geq 2, n \in N \). Hence or otherwise prove that \( \left|\begin{array}{ccc}3 & 1+s_{1} & 1+s_{2} \\ 1+s_{1} & 1+s_{2} & 1+s_{3} \\ 1+s_{2} & 1+s_{3} & 1+s_{4}\end{array}\right| \geq 0 \) for all real \( a, b, c \).
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