Question

Insert three harmonic means beween $\frac{1}{3}and\frac{1}{11}$.

• A. $\frac{1}{5},\frac{1}{7},\frac{1}{9}$
• B. $\frac{1}{7},\frac{1}{8},\frac{1}{8}$
• C. $4,6,8$
• D. $5,7,9$

Answer 1

Answer: (A)

$\frac{1}{5},\frac{1}{7},\frac{1}{9}$

Let $\frac{1}{3},h_1,h_2,h_3,\frac{1}{11}$

First finding $h_2$:

$\therefore h_2=\frac{2\times \frac{1}{3}\times \frac{1}{11}}{\frac{1}{3}+\frac{1}{11}}=\frac{1}{7}$

Now, finding H.M. between $\frac{1}{3}$ and $h_2$, i.e., $h_1$

$\therefore h_1=\frac{2\times \frac{1}{3}\times \frac{1}{7}}{\frac{1}{3}+\frac{1}{7}}=\frac{1}{5}$

Now, finding H.M. between $h_2$ and $\frac{1}{11}$, i.e., $h_3$

$\therefore h_3=\frac{2\times \frac{1}{7}\times \frac{1}{11}}{\frac{1}{7}+\frac{1}{11}}=\frac{1}{9}$

$\therefore$ Harmonic series: $\frac{1}{3},\frac{1}{5},\frac{1}{7},\frac{1}{9},\frac{1}{11}$