If - the roots of the equation x2-x-1-0 and An-n-n then An-2-An-2-

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Standard XI

Mathematics

Inequalities Involving Modulus Function

If α, the roo...

Question

$I f α, β are the roots of the equation x^{2} - x - 1 = 0 and A_{n} = α^{n} + β^{n} then A_{n + 2} + A_{n - 2} = - -$

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Solution

The correct option is B

$A_{n - 1} + A_{n - 2} = α^{n + 2} + β^{n + 2} + α^{n - 4} + β^{n - 4} = α^{n - 2} ((a^{2} + 1) - 2 a^{2}) + β^{n - 2} ((β^{2} + 1) - 2 β^{2}) (∵ α^{2} = α + 1) = α^{n - 2} ({(a + 2)}^{2} - 2 a^{2}) + β^{n - 2} ({(β + 2)}^{2} - 2 β^{2}) = 3 (α^{n} + β^{n}) = 3 A_{n}$

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If α,βare the roots of the equationx2−x−1=0andAn=αn+βnthenAn+2+An−2=−−

The correct option is B An−1+An−2=αn+2+βn+2+αn−4+βn−4=αn−2((a2+1)−2a2)+βn−2((β2+1)−2β2) (∵α2=α+1)=αn−2((a+2)2−2a2)+βn−2((β+2)2−2β2)=3(αn+βn)=3An

$I f α, β are the roots of the equation x^{2} - x - 1 = 0 and A_{n} = α^{n} + β^{n} then A_{n + 2} + A_{n - 2} = - -$