Paragraph: Let p, q be integers and let α, β be the roots of the equation, x^2-x-1=0 , where α ≠ β . For n=0,1,2, … , let a_n=p α^n+q β^n FACT: If a and b are rational numbers and a+b √5=0 , then a=0=b . Question: a_12=

Correct answer: (C) a_11+a_10. α^2=α+1 ⇒α^n=α^(n-1)+α^(n-2) ⇒ pα^n+qβ^n=pα^(n-1)+α^(n-2)+qβ^(n-1)+β^(n-2) a_n=a_(n-1)+a_(n-2) ⇒ a_12=a_11+a_10

Q36 / 54 · JEE Advanced 2017 (Paper 2) · Mathematics

Paragraph: Let $p, q$ be integers and let $\alpha, \beta$ be the roots of the equation, $x^{2}-x-1=0$, where $\alpha \neq \beta$. 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 $a=0=b$.
Question: $a_{12}=$

$α^{2} = α + 1 \Rightarrow α^{n} = α^{n - 1} + α^{n - 2}$

$\Rightarrow p α^{n} + q β^{n} = p (α^{n - 1} + α^{n - 2}) + q (β^{n - 1} + β^{n - 2})$

$a_{n} = a_{n - 1} + a_{n - 2}$

$\Rightarrow a_{12} = a_{11} + a_{10}$

Hence, the correct answer is option: 'C'.