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: a₁₂=

Step-by-step solution

α 2 = α + 1   ⇒ α n = α n - 1 + α n - 2 ⇒     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 ⇒     a 12 = a 11 + a 10