<p>Three non-zero numbers <em>a</em>, <em>b</em>, and <em>c</em> are in A.P. Increasing <em>a</em> by 1 or increasing <em>c</em> by 2, the numbers are in G.P. Then find <em>b</em>.</p>
Step-by-Step Solution
Key Concept: Set up two equations using the A.P. condition (2b = a + c) and two separate G.P. conditions (b² = (a+1)c and b² = a(c+2)), then solve by eliminating variables.
<p><strong>Step 1:</strong> From A.P. condition: 2b = a + c, so c = 2b - a</p><p><strong>Step 2:</strong> Case 1 - When a is increased by 1, numbers (a+1), b, c are in G.P.: b² = (a+1)c</p><p><strong>Step 3:</strong> Substitute c = 2b - a into b² = (a+1)c:</p><p>b² = (a+1)(2b - a)</p><p>b² = 2ab - a² + 2b - a</p><p>b² - 2ab + a² - 2b + a = 0</p><p>(b - a)² - 2b + a = 0 ... (i)</p><p><strong>Step 4:</strong> Case 2 - When c is increased by 2, numbers a, b, (c+2) are in G.P.: b² = a(c+2)</p><p><strong>Step 5:</strong> Substitute c = 2b - a into b² = a(c+2):</p><p>b² = a(2b - a + 2)</p><p>b² = 2ab - a² + 2a</p><p>b² - 2ab + a² - 2a = 0</p><p>(b - a)² - 2a = 0 ... (ii)</p><p><strong>Step 6:</strong> From equation (i): (b - a)² = 2b - a</p><p>From equation (ii): (b - a)² = 2a</p><p><strong>Step 7:</strong> Equating: 2b - a = 2a, therefore 2b = 3a</p><p>∴ <strong>b = 3a/2</strong></p>
Correct Answer: b = 3a/2