<p><strong>For Problems 10–12:</strong> Four different integers form an increasing A.P. One of these numbers is equal to the sum of the squares of the other three numbers.</p><p>The sum of all the four numbers is</p>
Step-by-Step Solution
Key Concept: Set up the four terms of an AP with a common difference, then use the condition that one term equals the sum of squares of the other three to form an equation. Solve for the common difference and first term.
<p><strong>Step 1:</strong> Let the four terms in increasing AP be: <em>a</em>, <em>a+d</em>, <em>a+2d</em>, <em>a+3d</em> where <em>d</em> > 0 (since they're increasing integers).</p><p><strong>Step 2:</strong> One number equals the sum of squares of the other three. Test the case where the fourth term equals the sum of squares of the first three:</p><p><em>a+3d = a² + (a+d)² + (a+2d)²</em></p><p><strong>Step 3:</strong> Expand the right side:</p><p><em>a+3d = a² + a² + 2ad + d² + a² + 4ad + 4d²</em></p><p><em>a+3d = 3a² + 6ad + 5d²</em></p><p><strong>Step 4:</strong> Rearrange:</p><p><em>3a² + 6ad - a + 5d² - 3d = 0</em></p><p><strong>Step 5:</strong> Treat as quadratic in <em>a</em>:</p><p><em>3a² + (6d-1)a + (5d²-3d) = 0</em></p><p><strong>Step 6:</strong> Using the quadratic formula and testing integer values of <em>d</em>:</p><p>For <em>d = 1</em>: <em>3a² + 5a + 2 = 0</em> → <em>(3a+2)(a+1) = 0</em> → <em>a = -1</em></p><p><strong>Step 7:</strong> The four terms are: <em>-1, 0, 1, 2</em></p><p>Verification: <em>2 = (-1)² + 0² + 1² = 1 + 0 + 1 = 2</em> ✓</p><p><strong>Step 8:</strong> Sum of all four numbers: <em>-1 + 0 + 1 + 2 = 2</em></p><p><strong>∴ Answer:</strong> D</p>
Correct Answer: D