If - - are the roots of the equation x2-ax-b-0 and An- n- n- then which of the following is true-

The correct option is B A_n+1=aA_n bA_n 1 x ^ 2 ax+b=0 #160;Roots are #160; α and β. x ^ 2 =ax b α #160; ^ 2 =aα b → (1) ...

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Quadratic Equations

If α, β are...

Question

If $α, β$ are the roots of the equation $x^{2} - a x + b = 0$ and $A_{n} = α^{n} + β^{n}$ , then which of the following is true?

A_{n + 1} = a A_{n} + b A_{n - 1}

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A_{n + 1} = b A_{n} + a A_{n - 1}

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A_{n + 1} = a A_{n} - b A_{n - 1}

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A_{n + 1} = b A_{n} - a A_{n - 1}

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α

β

x^{2} - a x + b = 0

A_{n} = α^{n} + β^{n}

a_{1} = \frac{1}{2} (a_{0} + \frac{A}{a_{0}}), a_{2} = \frac{1}{2} (a_{1} + \frac{A}{a_{1}}) a n d a_{n + 1} = \frac{1}{2} (a_{n} + \frac{A}{a_{n}})

\frac{a_{n} - \sqrt{A}}{a_{n} + \sqrt{A}} = (\frac{a_{1} - \sqrt{A}}{a_{1} + \sqrt{A}}) 2^{n - 1}

α

β

x^{2} - x - 1 = 0,

α

β

a_{n} = \frac{α^{n} - β^{n}}{α - β}, n \geq 1

b_{1} = 1

b_{n} = a_{n - 1} + a_{n + 1}, n \geq 2

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

α

β

x^{2} - a x + b = 0

A_{n} = α^{n} + β^{n}

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