Quadratic Equations
Roots and recurrence relations
Grade 11
Question:
<p>Let α and β be the roots of \(x^2 - 6x - 2 = 0\), with α > β. If \(a_n = \alpha^n - \beta^n\) for \(n \geq 1\), then the value of \(\dfrac{a_{10} - 2a_8}{2a_9}\) is</p>
<p>1</p>
<p>2</p>
<p>3</p>
<p>4</p>
Step-by-Step Solution
Key Concept: Use the recurrence relation satisfied by aₙ derived from the characteristic equation: aₙ = 6aₙ₋₁ + 2aₙ₋₂. This allows us to express a₁₀ and a₈ in terms of a₉ without computing actual root powers.
<p><strong>Step 1: Set up the recurrence relation</strong></p><p>Since α, β are roots of x² - 6x - 2 = 0:</p><p>• α + β = 6 and αβ = -2</p><p>• aₙ = αⁿ - βⁿ satisfies: <strong>aₙ = 6aₙ₋₁ + 2aₙ₋₂</strong></p><p><strong>Step 2: Express a₁₀ and a₈ in terms of a₉</strong></p><p>From the recurrence relation:</p><p>• a₁₀ = 6a₉ + 2a₈</p><p>• a₉ = 6a₈ + 2a₇, so a₈ = (a₉ - 2a₇)/6</p><p><strong>Step 3: Substitute into the target expression</strong></p><p>a₁₀ - 2a₈ = 6a₉ + 2a₈ - 2a₈ = 6a₉</p><p><strong>Step 4: Calculate the final answer</strong></p><p>$$\frac{a_{10} - 2a_8}{2a_9} = \frac{6a_9}{2a_9} = \frac{6}{2} = 3$$</p><p>∴ Answer: <strong>C (3)</strong></p>
Correct Answer: C