College Algebra, Chapter 9, 9.2, Section 9.2, Problem 58

Suppose the harmonic mean of two numbers is the reciprocal of the average of the reciprocals of the two numbers. Find the harmonic mean of 3 and 5.

Harmonic mean = $\displaystyle \frac{n}{\displaystyle \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \frac{1}{a_4} + ..... + \frac{1}{a_n}}$ where $n$ is a set of numbers

For the numbers 3 and 5,


$
\begin{equation}
\begin{aligned}

\text{harmonic mean } =& \frac{2}{\displaystyle \frac{1}{3} + \frac{1}{5}}
&& \text{Substitute } n = 2, a = 3 \text{ and } a_2 = 5
\\
\\
\text{harmonic mean } =& \frac{2}{\displaystyle \frac{8}{15}}
&& \text{Get the LCD}
\\
\\
\text{harmonic mean } =& \frac{30}{8}
&& \text{Multiply by its reciprocal}
\\
\\
\text{harmonic mean } =& \frac{15}{4}
&& \text{Simplify}

\end{aligned}
\end{equation}
$