Discussion
\(a, b, c\) are said to be in Harmonic Progression if their reciprocals \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) are in Arithmetic progression. What would be the value of \(x\) for which 3, \(x\), 6 are in Harmonic Progression.
Answer: B
The value of \(x\) is 4.
If 3, \(x\), 6 are in harmonic progression, their reciprocals are in arithmetic progression, so \(\frac{1}{x}\) is the average of \(\frac{1}{3}\) and \(\frac{1}{6}\):
\[ \begin{aligned} \frac{2}{x} &= \frac{1}{3} + \frac{1}{6} = \frac{1}{2} \\ x &= 4 \end{aligned} \]
Equivalently, the harmonic mean is \(\frac{2ab}{a+b} = \frac{36}{9} = 4\). 4½ is the arithmetic mean of 3 and 6, the trap; 5 and 5½ fit neither mean.
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