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

Solution

Maths sir · 4 steps

Factor the numerator in pairs, then let the root equation turn each difference into 4P.

1. Recurrence for P. Since α is a root, α² = α + 4. Multiply by αⁿ to get αⁿ⁺² = αⁿ⁺¹ + 4αⁿ. The same holds for β, and subtracting the two gives the result below.

   Pₙ₊₂ - Pₙ₊₁ = 4Pₙ

2. Factor the numerator. Group it as P₁₆(P₁₅ - P₁₄) - P₁₅(P₁₅ - P₁₄). This equals (P₁₅ - P₁₄)(P₁₆ - P₁₅).
3. Use the recurrence. With n = 13, P₁₅ - P₁₄ = 4P₁₃. With n = 14, P₁₆ - P₁₅ = 4P₁₄. So the numerator is 4P₁₃ × 4P₁₄ = 16P₁₃P₁₄.
4. So. Dividing by P₁₃P₁₄ leaves 16.

The idea to keep. When you see αⁿ ± βⁿ for roots of a quadratic, multiply the root equation by αⁿ to get a recurrence. Then look for differences of consecutive terms to factor out.

Answer from the NTA final answer key. Solution by PrepWiser.