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Relations between Roots and Coefficients

If α, β are t...

Question

If $α, β$ are the roots of the equation $a x^{2} + b x + c = 0 a n d S_{n} = α^{n} + β^{n} t h e n a S_{n + 1} + b S_{n} + c S_{n - 1} = (n \geq 2)$

A

$n^{2} a b c$

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B

$a + b + c$

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C

$(a + b + c) n$

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D

$0$

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Solution

The correct option is D
$0$

$S_{n + 1} = α^{n + 1} + β^{n + 1} = (α + β) (α^{n} + β^{n}) - α β (a^{n - 1} + β^{n - 1}) = - \frac{b}{a} . S_{n} - \frac{c}{a} . S_{n - 1}$

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