Sequences & Series
Newton's identities / power sums
Grade 11
Question:
<p>If \(\alpha, \beta, \gamma\) are such that \(\alpha + \beta + \gamma = 2\), \(\alpha^2 + \beta^2 + \gamma^2 = 6\), \(\alpha^3 + \beta^3 + \gamma^3 = 8\), then find the value of \(\alpha^4 + \beta^4 + \gamma^4\).</p>
Step-by-Step Solution
Key Concept: Use Newton's identities and power sum recurrence relations. Express higher power sums S_n = α^n + β^n + γ^n in terms of elementary symmetric polynomials (e₁, e₂, e₃) which can be determined from given conditions.
<p><strong>Step 1:</strong> Find elementary symmetric polynomials.</p><p>Let S_n = α^n + β^n + γ^n. Given: S₁ = 2, S₂ = 6, S₃ = 8</p><p>From (α + β + γ)² = α² + β² + γ² + 2(αβ + βγ + γα):</p><p>4 = 6 + 2e₂ ⟹ e₂ = αβ + βγ + γα = -1</p><p>where e₁ = α + β + γ = 2</p><p><strong>Step 2:</strong> Find e₃ using Newton's identity for S₃.</p><p>S₃ = e₁S₂ - e₂S₁ + 3e₃</p><p>8 = 2(6) - (-1)(2) + 3e₃</p><p>8 = 12 + 2 + 3e₃</p><p>3e₃ = -6 ⟹ e₃ = αβγ = -2</p><p><strong>Step 3:</strong> Apply Newton's identity for S₄.</p><p>S₄ = e₁S₃ - e₂S₂ + e₃S₁</p><p>S₄ = 2(8) - (-1)(6) + (-2)(2)</p><p>S₄ = 16 + 6 - 4</p><p>∴ Answer: <strong>18</strong></p>
Correct Answer: 18