Let α, β(α>β) be the roots of the quadratic equation x²-x-4=0. If P n =αⁿ-βⁿ, n N, then (P₁₅ P₁₆-P₁₄ P₁₆-P₁₅²+P₁₄ P₁₅)/(P₁₃ P₁₄) is equal to _ _ _ _ _.

Complex Numbers and Quadratic Equations: JEE Main 2022 Mathematics question

JEE Main 2022 · 29 Jul, Evening shift · Q43

JEE Main 202229 Jul, Evening shiftMathematicsMediumComplex Numbers and Quadratic Equations264× since 2002

Let

\alpha, \beta(\alpha>\beta)

be the roots of the quadratic equation

x^{2}-x-4=0 .

P_{n}=\alpha^{n}-\beta^{n}

n \in \mathrm{N}

, then

\frac{P_{15} P_{16}-P_{14} P_{16}-P_{15}^{2}+P_{14} P_{15}}{P_{13} P_{14}}

is equal to __________.

Your answerType your answer. Integers and decimals are accepted.