Quadratic Equations
Roots and Sum of Powers
Grade 11
Question:
<p>If <span>α</span>, <span>β</span> are the roots of the equation <span>ax<sup>2</sup> + bx + c = 0</span> and <span>A<sub>n</sub> = α<sup>n</sup> + β<sup>n</sup></span>, then <span>aA<sub>n+2</sub> + bA<sub>n+1</sub> + cA<sub>n</sub></span> is equal to</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) a + b + c</p>
<p>(d) abc</p>
Step-by-Step Solution
Key Concept: Establish the recurrence relation for sums of powers of roots using the fact that α and β satisfy the original quadratic equation.
<p><strong>Step 1:</strong> Since α and β are roots of <span>ax<sup>2</sup> + bx + c = 0</span>, we have:</p><p><span>α + β = -\frac{b}{a}</span> and <span>αβ = \frac{c}{a}</span></p><p><strong>Step 2:</strong> <span>A<sub>n+2</sub> = α<sup>n+2</sup> + β<sup>n+2</sup> = (α + β)(α<sup>n+1</sup> + β<sup>n+1</sup>) - αβ(α<sup>n</sup> + β<sup>n</sup>)</span></p><p><span>A<sub>n+2</sub> = (α + β)A<sub>n+1</sub> - αβA<sub>n</sub></span></p><p><strong>Step 3:</strong> <span>A<sub>n+2</sub> = -\frac{b}{a}A<sub>n+1</sub> - \frac{c}{a}A<sub>n</sub></span></p><p><strong>Step 4:</strong> Multiplying by a: <span>aA<sub>n+2</sub> = -bA<sub>n+1</sub> - cA<sub>n</sub></span></p><p><strong>Step 5:</strong> Therefore, <span>aA<sub>n+2</sub> + bA<sub>n+1</sub> + cA<sub>n</sub> = 0</span></p><p>∴ Answer is (a).</p>
Correct Answer: A