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=

Question

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= (A) 21 (B) 14 (C) 7 (D) 12

Tardigrade Admin · Accepted Answer

a 4=2 a 0+3 a 1 a 4=( q - p )(3 β)+5 p +2 q =28 ⇒ p = q =4

Q. 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,…, let an​= pαn+qβn FACT : If a and b are rational numbers and a+b5​=0, then a=0=b. If a4​=28, then p+2q=

21

14

7

12

a4​=2a0​+3a1​ a4​=(q−p)(3β)+5p+2q=28 ⇒p=q=4

$a_{4} = 2 a_{0} + 3 a_{1}$
$a_{4} = (q - p) (3 β) + 5 p + 2 q = 28$
$\Rightarrow p = q = 4$