Paragraph: Let p, q be integers and let , be the roots of the equation, x²-x-1=0 , where . For n=0,1,2, , let a_ n =p ^ n +q ^ n FACT: If a and b are rational numbers and a+b 5 =0 , then a=0=b . Question: If a₄=28 , then p+2 q=

Step-by-step solution

α 2 = α + 1 ⇒ α 4 = ( α 2 ) 2 ⇒ α 4 = ( α + 1 ) 2 ⇒ α 4 = α 2 + 2 α + 1 ⇒ α 4 = α + 1 + 2 α + 1 ⇒ α 4 = 3 α + 2 A l s o   α + β = 1 ∴         a 4 = 28   ⇒     p α 4 + q β 4 = p 3 α + 2 + q 3 β + 2 = 2 8 ⇒       p 3 α + 2 + q 3 - 3 α + 2 = 2 8 ⇒       α 3 p - 3 q + 2 p + 5 q =