Sequences & Series
Harmonic Means
Grade 11
Question:
<p>Suppose <math>q</math> is the first of <math>n</math> harmonic means between two positive numbers <math>a</math> and <math>b</math>. The value of <math>q</math> is</p>
<p>(a) <math>\frac{(n+1)ab}{nb + a}</math></p>
<p>(b) <math>\frac{(n+1)ab}{na + b}</math></p>
<p>(c) <math>\frac{(n-1)ab}{na + b}</math></p>
<p>(d) <math>\frac{(n-1)ab}{nb + a}</math></p>
Step-by-Step Solution
Key Concept: Harmonic means are defined through the arithmetic progression of their reciprocals.
<p>Harmonic means correspond to the reciprocals forming an AP. If <math>q</math> is the first harmonic mean between <math>a</math> and <math>b</math>, then <math>\frac{1}{a}, \frac{1}{q}, \ldots, \frac{1}{b}</math> form an AP. The first term of this AP of reciprocals is <math>\frac{1}{q} = \frac{1}{a} + \frac{b-a}{(n+1)ab}</math>. Solving for <math>q</math> gives <math>q = \frac{(n+1)ab}{nb + a}</math>.</p>
Correct Answer: A