Let α, β be the roots of the equation x²-ax-b=0 with Im(α)<Im(β). Let Pₙ=αⁿ-βⁿ. If P₃=-5 √7 i, P₄=-3 √7 i, P₅=11 √7 i and P₆=45 √7 i, then |α⁴+β⁴| is equal to __________.

Solution

Maths sir · 4 steps

Each root satisfies x² = ax + b, so the Pₙ follow a recurrence. Two given values pin down a and b.

1. Build the recurrence. Since α² = aα + b, multiply by αⁿ to get αⁿ⁺² = aαⁿ⁺¹ + bαⁿ. The same holds for β. Subtract the two to get a recurrence for Pₙ.

   Pₙ₊₂ = a Pₙ₊₁ + b Pₙ

2. Find a and b. Cancel √7 i everywhere. With n = 3: 11 = -3a - 5b. With n = 4: 45 = 11a - 3b. Solving these gives a = 3 and b = -4. The equation is x² - 3x + 4 = 0.
3. Sum and product. α + β = 3 and αβ = 4. So α² + β² = 3² - 2 × 4 = 1.
4. So. α⁴ + β⁴ = (α² + β²)² - 2(αβ)² = 1 - 32 = -31. Its modulus is 31.

The common slip: 257. Taking the product of roots as b = -4 instead of -b = 4 gives α² + β² = 17 and then 257, not 31.

The idea to keep. For any power sum or difference of roots of x² = ax + b, use Sₙ₊₂ = aSₙ₊₁ + bSₙ. Then get α⁴ + β⁴ from the sum and product of the roots.

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