<p><strong>25.</strong> If \(a, b\) and \(c\) are in A.P. and \(b - a, c - b\) and \(a\) are in G.P., then \(a:b:c\) is</p>
Step-by-Step Solution
Key Concept: Since a, b, c are in A.P., we have b - a = c - b (common difference). Combined with the G.P. condition on (b-a), (c-b), a, we can set up equations using the G.P. property: (c-b)² = (b-a)·a, which yields a specific ratio.
<p><strong>Step 1:</strong> Since a, b, c are in A.P., we have:</p><p>b - a = c - b</p><p>Let common difference = d, so b = a + d and c = a + 2d</p><p><strong>Step 2:</strong> Since (b-a), (c-b), a are in G.P., the common ratio property gives:</p><p>(c - b)² = (b - a) · a</p><p>d² = d · a</p><p>Since d ≠ 0 (otherwise all terms are equal), we get: d = a</p><p><strong>Step 3:</strong> Substitute d = a back:</p><p>b = a + a = 2a</p><p>c = a + 2a = 3a</p><p><strong>Step 4:</strong> Verify G.P. condition: (b-a) = a, (c-b) = a, common ratio = a/a = 1 ✓</p><p>∴ Answer: a : b : c = <strong>1 : 2 : 3</strong></p>
Correct Answer: A