Question

Insert three harmonic means between 9 and 13.

• A. $11,12,13$
• B. $\frac{12}{117},\frac{11}{117},\frac{10}{117}$
• C. $\frac{13}{9},\frac{13}{7},\frac{13}{6}$
• D. $\frac{117}{12},\frac{117}{11},\frac{117}{10}$

Answer 1

The correct option is D $\frac{117}{12},\frac{117}{11},\frac{117}{10}$

Let $9,h1,h2,h3,13$

First finding $h2$:

$\therefore h2=\frac{2\times 9\times 13}{9+13}=\frac{117}{11}$

Now, finding H.M. between $9$ and $h2$, i.e., $h1$

$\therefore h1=\frac{2\times 9\times \frac{117}{11}}{9+\frac{117}{11}}=\frac{117}{12}$

Now, finding H.M. between $h2$ and $13$, i.e., $h3$

$\therefore h3=\frac{2\times \frac{117}{11}\times 13}{\frac{117}{11}+13}=\frac{117}{10}$

$\therefore$ Harmonic series: $9,\frac{117}{12},\frac{117}{11},\frac{117}{10},13$