Let alpha and beta be the roots of the equati | vedclass

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(A) Since $\alpha$ and $\beta$ are roots of $5x^{2} + 6x - 2 = 0$,they satisfy the equation.$5\alpha^{2} + 6\alpha - 2 = 0 \implies 5\alpha^{n+2} + 6\

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$5S_{6} + 6S_{5} = 2S_{4}$

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(A) Since $\alpha$ and $\beta$ are roots of $5x^{2} + 6x - 2 = 0$,they satisfy the equation.$5\alpha^{2} + 6\alpha - 2 = 0 \implies 5\alpha^{n+2} + 6\alpha^{n+1} - 2\alpha^{n} = 0$ (multiplying by $\alpha^{n}$).Similarly,$5\beta^{n+2} + 6\beta^{n+1} - 2\beta^{n} = 0$.Adding these two equations,we get $5(\alpha^{n+2} + \beta^{n+2}) + 6(\alpha^{n+1} + \beta^{n+1}) - 2(\alpha^{n} + \beta^{n}) = 0$.This simplifies to $5S_{n+2} + 6S_{n+1} - 2S_{n} = 0$.For $n = 4$,we have $5S_{6} + 6S_{5} - 2S_{4} = 0$,which implies $5S_{6} + 6S_{5} = 2S_{4}$.

Let $\alpha$ and $\beta$ be the roots of the equation $5x^{2} + 6x - 2 = 0$. If $S_{n} = \alpha^{n} + \beta^{n}$ for $n = 1, 2, 3, \dots$,then:

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