Sequences & Series
Harmonic Mean
Grade 11
Question:
<p>318. If \(H\) be the harmonic mean between \(P\) and \(Q\), then the value of \(\frac{H}{P} + \frac{H}{Q}\) is</p>
<p>(a) 2</p>
<p>(b) \(\frac{PQ}{P+Q}\)</p>
<p>(c) \(\frac{P+Q}{PQ}\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: The harmonic mean H of two numbers P and Q is defined as H = 2PQ/(P+Q). Use this definition to substitute and simplify the given expression algebraically.
<p><strong>Step 1:</strong> Recall that if H is the harmonic mean between P and Q, then:</p><p>H = 2PQ/(P+Q)</p><p><strong>Step 2:</strong> Calculate H/P:</p><p>H/P = [2PQ/(P+Q)]/P = 2Q/(P+Q)</p><p><strong>Step 3:</strong> Calculate H/Q:</p><p>H/Q = [2PQ/(P+Q)]/Q = 2P/(P+Q)</p><p><strong>Step 4:</strong> Add the two expressions:</p><p>H/P + H/Q = 2Q/(P+Q) + 2P/(P+Q) = [2Q + 2P]/(P+Q) = 2(P+Q)/(P+Q) = 2</p><p>∴ Answer: <strong>2</strong></p>
Correct Answer: A