<p>If <em>a</em>, <em>b</em>, <em>c</em>, <em>d</em> are distinct integers in an A.P. such that \(d = a^2 + b^2 + c^2\), then find the value of \(a + b + c + d\).</p>
Step-by-Step Solution
Key Concept: Since a, b, c, d are in A.P., express them as a, a+r, a+2r, a+3r. Then use the constraint d = a² + b² + c² to form an equation in terms of a and r, which severely limits possible integer solutions.
<p><strong>Step 1:</strong> Let the four terms in A.P. be: a, a+r, a+2r, a+3r where r is the common difference.</p><p><strong>Step 2:</strong> So b = a+r, c = a+2r, d = a+3r.</p><p><strong>Step 3:</strong> Using the constraint d = a² + b² + c²:</p><p>a+3r = a² + (a+r)² + (a+2r)²</p><p>a+3r = a² + a² + 2ar + r² + a² + 4ar + 4r²</p><p>a+3r = 3a² + 6ar + 5r²</p><p><strong>Step 4:</strong> Rearranging: 3a² + 6ar - a + 5r² - 3r = 0</p><p>This is a quadratic in a: 3a² + a(6r - 1) + (5r² - 3r) = 0</p><p><strong>Step 5:</strong> For integer solutions, the discriminant must be a perfect square:</p><p>Δ = (6r-1)² - 12(5r² - 3r) = 36r² - 12r + 1 - 60r² + 36r = -24r² + 24r + 1</p><p><strong>Step 6:</strong> Testing small integer values of r:</p><p>• r = 1: Δ = -24 + 24 + 1 = 1 = 1² ✓</p><p>• r = -1: Δ = -24 - 24 + 1 = -47 ✗</p><p>• r = 2: Δ = -96 + 48 + 1 = -47 ✗</p><p><strong>Step 7:</strong> For r = 1: 3a² + 5a + 2 = 0 → (3a + 2)(a + 1) = 0</p><p>So a = -1 or a = -2/3 (not integer)</p><p><strong>Step 8:</strong> With a = -1, r = 1: The sequence is -1, 0, 1, 2</p><p>Check: d = 2 and a² + b² + c² = 1 + 0 + 1 = 2 ✓</p><p><strong>Step 9:</strong> Therefore: a + b + c + d = -1 + 0 + 1 + 2 = <strong>2</strong></p>
Correct Answer: 2