Question

Insert three harmonic means between $\frac{1}{5},\frac{1}{17}$.

• A. $8,11,14$
• B. $\frac{1}{8},\frac{1}{11},\frac{1}{14}$
• C. $\frac{8}{85},\frac{11}{85},\frac{14}{85}$
• D. $\frac{85}{8},\frac{85}{11},\frac{85}{14}$

Answer 1

The correct option is B $\frac{1}{8},\frac{1}{11},\frac{1}{14}$

Let $\frac{1}{5},h1,h2,h3,\frac{1}{17}$

First finding $h2$:

$\therefore h2=\frac{2\times \frac{1}{5}\times \frac{1}{17}}{\frac{1}{5}+\frac{1}{17}}=\frac{1}{11}$

Now, finding H.M. between $\frac{1}{5}$ and $h2$, i.e., $h1$

$\therefore h1=\frac{2\times \frac{1}{5}\times \frac{1}{11}}{\frac{1}{5}+\frac{1}{11}}=\frac{1}{8}$

Now, finding H.M. between $h2$ and $\frac{1}{17}$, i.e., $h3$

$\therefore h3=\frac{2\times \frac{1}{11}\times \frac{1}{17}}{\frac{1}{11}+\frac{1}{17}}=\frac{1}{14}$

$\therefore$ Harmonic series: $\frac{1}{5},\frac{1}{8},\frac{1}{11},\frac{1}{14},\frac{1}{17}$