JEE Advanced 2017 · Paper 2 · Mathematics Q17

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MathematicsSingle correct+3Question 53 of 54

Passage

PARAGRAPH 2

Let $p$ , $q$ be integers and let $\alpha$ , $\beta$ be the roots of the equation, $x^2 - x - 1 = 0$ , where $\alpha \ne \beta$ . For $n = 0, 1, 2, \ldots$ , let $a_n = p\alpha^n + q\beta^n$ .

FACT: If $a$ and $b$ are rational numbers and $a + b\sqrt{5} = 0$ , then $a = 0 = b$ .

$a_{12} =$

a_{11} - a_{10}

a_{11} + a_{10}

2a_{11} + a_{10}

a_{11} + 2a_{10}

Answer and worked solution

Official answer: B

Since $\alpha^2=\alpha+1$ , $\alpha^{n+2}=\alpha^{n+1}+\alpha^n$ , and the same holds for $\beta$ . So
$a_{n+2}=a_{n+1}+a_n\implies a_{12}=a_{11}+a_{10}.$

(B)

Topics:Roots and coefficients Special series sums

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JEE Advanced 2017 · Paper 2, Mathematics question 17