<p>If \(a, b, c, d\) are distinct integers in AP such that \(d = a^2 + b^2 + c^2\), then find \(\dfrac{(a+b+c+d)}{5}\).</p>
Step-by-Step Solution
Key Concept: Since a, b, c, d are in AP with common difference r, express them as a, a+r, a+2r, a+3r. The constraint d = a² + b² + c² becomes a cubic equation in terms of a and r, which has limited integer solutions due to the distinctness requirement.
<p><strong>Step 1:</strong> Express terms in AP as: a, a+r, a+2r, a+3r where r is common difference and a, b, c, d are distinct integers.</p><p><strong>Step 2:</strong> Apply 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 3:</strong> Rearrange: 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 4:</strong> For integer solutions, 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 5:</strong> Test small integer values of r:</p><p>r = 1: Δ = -24 + 24 + 1 = 1 = 1² ✓</p><p>a = [-(6-1) ± 1]/6 = [-5 ± 1]/6 → a = -2/3 (not integer) or a = -1</p><p><strong>Step 6:</strong> With a = -1, r = 1: a = -1, b = 0, c = 1, d = 1 + 0 + 1 = 2</p><p>Check: -1, 0, 1, 2 are in AP ✓ and distinct ✓</p><p><strong>Step 7:</strong> Calculate:</p><p>(a + b + c + d)/5 = (-1 + 0 + 1 + 2)/5 = 2/5</p><p>∴ Answer: <strong>2/5</strong></p>
Correct Answer: 2