Sequences & Series
Harmonic Progression
Grade 11
Question:
<p>If <i>A</i>, <i>G</i>, <i>H</i> be respectively the A.M., G.M. and H.M. between two positive numbers and if <i>xA</i> = <i>yG</i> = <i>zH</i> where <i>x</i>, <i>y</i>, <i>z</i> are non-zero positive quantities, then <i>x</i>, <i>y</i>, <i>z</i> are in</p>
<p>(A) A.P.</p>
<p>(B) G.P.</p>
<p>(C) H.P.</p>
<p>(D) Nothing can be said</p>
Step-by-Step Solution
Key Concept: Use the relationship between A.M., G.M., and H.M. along with the property that reciprocals of terms in H.P. form an A.P.
<p>Let the common value be <i>k</i>, so <i>xA</i> = <i>yG</i> = <i>zH</i> = <i>k</i>.</p><p>Then <i>x</i> = <i>k</i>/<i>A</i>, <i>y</i> = <i>k</i>/<i>G</i>, <i>z</i> = <i>k</i>/<i>H</i>.</p><p>Since <i>A</i> ≥ <i>G</i> ≥ <i>H</i> for positive numbers, we have <i>x</i> ≤ <i>y</i> ≤ <i>z</i>.</p><p>By the AM-GM-HM relationship: <i>1/x</i>, <i>1/y</i>, <i>1/z</i> are in A.P., which means <i>x</i>, <i>y</i>, <i>z</i> are in H.P.</p><p>∴ Answer is C.</p>
Correct Answer: C