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 a

Question

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 ​ =

TestHub · Accepted Answer

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

Mathematics - Quadratic Equation Question with Solution | TestHub

Options:

Doubts & Discussion