If α, β are roots of the equation x² + 5(2)x + 10 = 0, α > β and P n = α ⁿ - β ⁿ for each positive integer n, then the value of ((P₁₇P₂₀ + 5 2 P₁₇P₁₉)/(P₁₈P₁₉ + 5 2 P₁₈²)) is equal to _ _ _ _ _.

Complex Numbers and Quadratic Equations: JEE Main 2021 Mathematics question

JEE Main 2021 · 25 Jul, Morning shift · Q43

JEE Main 202125 Jul, Morning shiftMathematicsMediumComplex Numbers and Quadratic Equations264× since 2002

\alpha

\beta

are roots of the equation

{x^2} + 5(\sqrt 2 )x + 10 = 0

\alpha

\beta

and

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

for each positive integer n, then the value of

\left( {{{{P_{17}}{P_{20}} + 5\sqrt 2 {P_{17}}{P_{19}}} \over {{P_{18}}{P_{19}} + 5\sqrt 2 P_{18}^2}}} \right)

is equal to _________.

Your answerType your answer. Integers and decimals are accepted.