Consider integers p and q, and let α and β be the roots of the quadratic equation x² - x - 1 = 0, where α ≠ β. Define a sequence aₙ = pαⁿ + qβⁿ for n = 0, 1, 2,.... Given that a₄ = 28 and the property that if a and b are rational numbers such that a + b√5 = 0, then both a and b must equal

Question

Consider integers p and q, and let α and β be the roots of the quadratic equation x² - x - 1 = 0, where α ≠ β. Define a sequence aₙ = pαⁿ + qβⁿ for n = 0, 1, 2,.... Given that a₄ = 28 and the property that if a and b are rational numbers such that a + b√5 = 0, then both a and b must equal zero, what is the value of p + 2q?

Accepted Answer

12. The value of p + 2q is 12, which can be determined by using the given quadratic equation and the properties of its roots α and β to derive the relationship between p and q from the equation a₄ = 28.

Solution

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