Sequences & Series
G.P. and Inequalities
Grade 11
Question:
<p>The sum of squares of 3 distinct real numbers which are in G.P. is \(s^2\). If their sum is \(\alpha s\), then \(\alpha^2\) lies in interval</p>
<p>(A) \(\left(\frac{1}{3}, 1\right)\)</p>
<p>(B) \((0, 3)\)</p>
<p>(C) (1, 3)</p>
<p>(D) (3, 9)</p>
Step-by-Step Solution
Key Concept: Use G.P. terms and relate sum of squares to sum using algebraic manipulation.
<p><strong>Let the three terms be:</strong> \(\frac{a}{r}, a, ar\) where \(r \neq 1\)</p><p><strong>Sum of squares:</strong> \(\frac{a^2}{r^2} + a^2 + a^2r^2 = s^2\)</p><p><strong>Sum:</strong> \(\frac{a}{r} + a + ar = \alpha s\)</p><p><strong>After squaring and simplification:</strong> \(1 < \alpha^2 < 3\)</p>
Correct Answer: C