<p>If x, 2y and 3z are in AP, where the distinct numbers x, y, z are in GP, then the common ratio of the GP is</p>
Step-by-Step Solution
Key Concept: Since x, y, z are in GP, we can express them as x, xr, xr² where r is the common ratio. The condition that x, 2y, 3z are in AP gives us a quadratic equation in r, which we solve to find the specific value.
<p><strong>Step 1:</strong> Since x, y, z are in GP with common ratio r, write them as:</p><p>x, xr, xr²</p><p><strong>Step 2:</strong> The condition states that x, 2y, 3z are in AP. For three terms to be in AP, the middle term must equal the average of the first and third:</p><p>2(2y) = x + 3z</p><p>4y = x + 3z</p><p><strong>Step 3:</strong> Substitute y = xr and z = xr²:</p><p>4(xr) = x + 3(xr²)</p><p>4xr = x + 3xr²</p><p><strong>Step 4:</strong> Divide both sides by x (since x ≠ 0):</p><p>4r = 1 + 3r²</p><p>3r² - 4r + 1 = 0</p><p><strong>Step 5:</strong> Factor the quadratic:</p><p>3r² - 4r + 1 = (3r - 1)(r - 1) = 0</p><p><strong>Step 6:</strong> This gives r = 1/3 or r = 1</p><p><strong>Step 7:</strong> Since x, y, z are distinct numbers, r ≠ 1. Therefore, r = 1/3.</p><p><strong>Verification:</strong> If r = 1/3, then y = x/3 and z = x/9. Check: x, 2(x/3), 3(x/9) = x, 2x/3, x/3. Differences: 2x/3 - x = -x/3 and x/3 - 2x/3 = -x/3 ✓ (These are equal, confirming AP)</p><p><strong>∴ Answer:</strong> b</p>
Correct Answer: b