Question

Insert three harmonic means between $1$ and $5.$

• A. $\frac{1}{4},\frac{1}{3},\frac{1}{2}$
• B. $\frac{5}{4},\frac{5}{3},\frac{5}{2}$
• C. $\frac{4}{5},\frac{3}{5},\frac{2}{5}$
• D. $2,3,4$

Answer 1

The correct option is B $\frac{5}{4},\frac{5}{3},\frac{5}{2}$

Let $1,h1,h2,h3,5$

First finding $h2$:

$\therefore h2=\frac{2\times 5\times 1}{5+1}=\frac{5}{3}$

Now, finding H.M. between $1$ and $h2$, i.e., $h1$

$\therefore h1=\frac{2\times 1\times \frac{5}{3}}{1+\frac{5}{3}}=\frac{5}{4}$

Now, finding H.M. between $h2$ and $5$, i.e., $h3$

$\therefore h3=\frac{2\times \frac{5}{3}\times 5}{\frac{5}{3}+5}=\frac{5}{2}$

$\therefore$ Harmonic series: $1,\frac{5}{4},\frac{5}{3},\frac{5}{2},5$