Sequences & Series
Progressions
Grade 11
Question:
<p>If <i>a</i>, <i>b</i>, <i>c</i> be in A.P. and \(a^2, b^2, c^2\) be in H.P., then which of the following(s) may be true?</p>
<p>(A) <i>a</i>, <i>b</i>, <i>c</i> are in G.P.</p>
<p>(B) <i>a</i>, [?], <i>c</i> are in G.P.</p>
<p>(C) [?], <i>b</i>, <i>c</i> are in G.P.</p>
<p>(D) <i>a</i> = <i>b</i> = <i>c</i></p>
Step-by-Step Solution
Key Concept: Use the A.P. condition and H.P. condition simultaneously to determine valid relationships between a, b, c.
<p><strong>Step 1:</strong> <i>a</i>, <i>b</i>, <i>c</i> in A.P. ⟹ \(2b = a + c\)<br/><strong>Step 2:</strong> \(a^2, b^2, c^2\) in H.P. ⟹ \(\frac{2}{b^2} = \frac{1}{a^2} + \frac{1}{c^2} = \frac{a^2+c^2}{a^2c^2}\)<br/>\(2a^2c^2 = b^2(a^2+c^2)\)<br/><strong>Step 3:</strong> From A.P.: \(c = 2b - a\). Substitute and simplify.<br/>\(2a^2(2b-a)^2 = b^2[a^2 + (2b-a)^2]\)<br/>\(2a^2(4b^2 - 4ab + a^2) = b^2(a^2 + 4b^2 - 4ab + a^2)\)<br/>\(8a^2b^2 - 8a^3b + 2a^4 = 2a^2b^2 + 4b^4 - 4ab^3\)<br/><strong>Step 4:</strong> Testing <i>a</i> = <i>b</i> = <i>c</i>: Both conditions satisfied. ∴ Answer is D.</p>
Correct Answer: D