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

Solution

Maths sir · 4 steps

Each root satisfies the quadratic, so Pₙ obeys a recurrence. Factor first, then use it twice.

1. Build the recurrence. Since α is a root, α² + 5√2 α + 10 = 0. Multiply by αⁿ, do the same for β, and subtract. The powers combine into P terms.

   Pₙ₊₂ + 5√2 Pₙ₊₁ + 10 Pₙ = 0 ⇒ Pₙ₊₂ + 5√2 Pₙ₊₁ = -10 Pₙ

2. Numerator. Factor out P₁₇ to get P₁₇(P₂₀ + 5√2 P₁₉). With n = 18, the bracket is -10P₁₈, so the numerator is -10P₁₇P₁₈.
3. Denominator. Factor out P₁₈ to get P₁₈(P₁₉ + 5√2 P₁₈). With n = 17, the bracket is -10P₁₇, so the denominator is -10P₁₇P₁₈.
4. So. The discriminant is 50 - 40 = 10 > 0, so the roots are real and distinct with |α| ≠ |β|. That means no Pₙ is zero. The ratio is -10P₁₇P₁₈/-10P₁₇P₁₈ = 1.

The idea to keep. When you see αⁿ ± βⁿ with large n, never compute the roots. Turn the quadratic into a recurrence Pₙ₊₂ + bPₙ₊₁ + cPₙ = 0 and look for brackets that match it.

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