JEE Main 2022 · 29 Jul · Shift 2 · Mathematics Q22

← JEE Main 2022 · 29 Jul · Shift 2

MathematicsInteger+4 / -1Question 22 of 90

Let α, β (α > β) be the roots of the quadratic equation $x^2-x-4=0$ . If $P_n=\alpha^n-\beta^n$ , n ∈ ℕ, then $\dfrac{P_{15}P_{16}-P_{14}P_{16}-P_{15}^2+P_{14}P_{15}}{P_{13}P_{14}}$ is equal to _______.

Your answer

Answer and worked solution

Official answer: 16

Recurrence. $\alpha^2=\alpha+4$ gives $P_{n+1}=P_n+4P_{n-1}$ .

Factor the numerator as $(P_{15}-P_{14})(P_{16}-P_{15})=4P_{13}\cdot4P_{14}$ . Ratio 16.

Answer: 16

Topics:Roots and coefficients Special series sums

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JEE Main 2022 · 29 Jul · Shift 2, Mathematics question 22