1. Tardigrade
  2. Question
  3. Mathematics
  4. PARAGRAPH 2 Let p, q be integers and let α, β be the roots of the equation, x2-x-1=0, where α ≠ β. For n=0,1,2, ldots, let an= p αn+q βn FACT: If a and b are rational numbers and a+b √5=0, then a=0=b. If a4=28, then p+2 q=

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Q. PARAGRAPH 2

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 $a=0=b$.

If $a_{4}=28$, then $p+2 q=$

JEE AdvancedJEE Advanced 2017Complex Numbers and Quadratic Equations

Solution:

$a _{4}=2 a _{0}+3 a _{1}$
$a _{4}=( q - p )(3 \beta)+5 p +2 q =28$
$\Rightarrow p = q =4$