Let α and β be the roots of the equation x² - x - 1 = 0.
If pₖ = ( α )ᵏ + ( β )ᵏ , k ≥ 1, then which one of the following statements is not true?

Solution

Maths sir · 3 steps

Each root satisfies x² = x + 1, so every pₖ is the sum of the two before it.

1. Build the recurrence. Since α² = α + 1, multiplying by αᵏ gives αᵏ⁺² = αᵏ⁺¹ + αᵏ. The same holds for β. Add the two and you get the recurrence below.

   pₖ₊₂ = pₖ₊₁ + pₖ

2. Starting values. p₁ = α + β = 1, and p₂ = (α + β)² - 2αβ = 1 - 2(-1) = 3. Then p₃ = 4, p₄ = 7, p₅ = 11.

   Option	Check	True?
   (a)	1 + 3 + 4 + 7 + 11 = 26	true
   (b)	p₅ = 11	true
   (c)	p₅ - p₄ = 11 - 7 = 4 = p₃	true
   (d)	p₂ · p₃ = 3 × 4 = 12 ≠ 11	false

3. So. The statement that is not true is (d), because p₅ = 11 but p₂ · p₃ = 12.

The idea to keep. If α, β are roots of x² = ax + b, then the power sums obey pₖ₊₂ = a pₖ₊₁ + b pₖ. Find p₁ and p₂, then climb up.

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