Sequences & Series
Means
Grade 11
Question:
<p>Let <span style='font-style:italic'>p</span> be the first of <span style='font-style:italic'>n</span> arithmetic means between two positive numbers and <span style='font-style:italic'>q</span> be first of <span style='font-style:italic'>n</span> harmonic means between same two numbers. Then \(\frac{p}{q}\) can lie in interval(s)</p>
<p>(A) \((-\infty, 1]\)</p>
<p>(B) \(\left[1, \frac{n+1}{n}\right)\)</p>
<p>(C) \(\left(\frac{n}{n+1}, 1\right)\)</p>
<p>(D) \(\left(1, \frac{n+1}{n}\right)\)</p>
Step-by-Step Solution
Key Concept: Use relationship between arithmetic and harmonic means, combined with AM-HM inequality.
<p><strong>For two positive numbers A and B:</strong></p><p><strong>First A.M.:</strong> \(p = A + \frac{B-A}{n+1}\)</p><p><strong>First H.M.:</strong> \(q = \frac{2AB}{A+B} \cdot \frac{n+1}{n+1}\)</p><p><strong>By AM-HM inequality:</strong> \(p \geq q\) with ratio dependent on n and position.</p><p>Therefore \(\frac{p}{q} \in \left(1, \frac{n+1}{n}\right)\)</p>
Correct Answer: D