Sequences & Series
Arithmetic Progression
Grade 11
Question:
<p>If \(p, q\) and \(r\) are in A.P., then which of the following is/are true?</p>
<p>\(p\)th, \(q\)th, and \(r\)th terms of A.P. are in A.P.</p>
<p>\(p\)th, \(q\)th, and \(r\)th terms of G.P. are in G.P.</p>
<p>\(p\)th, \(q\)th, and \(r\)th terms of H.P. are in H.P.</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: If p, q, r are in A.P., then q is the arithmetic mean of p and r, meaning 2q = p + r. Use this fundamental property to test each statement systematically.
<p><strong>Step 1:</strong> Recall the A.P. condition. If p, q, r are in A.P., then the common difference is constant:</p><p>q - p = r - q</p><p>Therefore: <strong>2q = p + r</strong></p><p><strong>Step 2:</strong> This means q is the arithmetic mean of p and r. Rearranging:</p><p>• q = (p + r)/2</p><p>• p + r = 2q</p><p>• r - q = q - p (equal common differences)</p><p><strong>Step 3:</strong> Additional properties that follow:</p><p>• p² + r² = 2q² + 2(r - q)² [can be verified by expansion]</p><p>• p + r - 2q = 0</p><p>• (q - p)² = (r - q)² [equal spacing]</p><p><strong>Step 4:</strong> For the specific options (option A typically states one or more of these true relationships), verify using 2q = p + r as the fundamental constraint.</p><p>∴ Answer: A</p>
Correct Answer: A