Question

Insert six harmonic means between 3 and 6/23.Find $22$ times of the third term of the six terms ?

Answer 1

Inserting 6 harmonic means between $3$ and $\frac{6}{23}$ is equivalent in inserting 6 arithmetic means between $\frac{1}{3}$ and $\frac{23}{6}$.
We make the denominators equal to $6$, thus giving the first and last numerators as $2$ and $23$
Inserting 6 arithmetic means between $2,23$ will give the common difference as $\frac{23-2}{7}$ since there will be $7$ gaps.
Thus, we get the terms as $2,5,8,11,14,17,20,23$
Hence the actual terms in A.P. are $\frac{2}{6},\frac{5}{6},\frac{8}{6},\frac{11}{6},\frac{14}{6},\frac{17}{6},\frac{20}{6},\frac{23}{6}$
Also, terms in H.P. look like $3,\frac{6}{5},\frac{6}{8},\frac{6}{11},\frac{6}{14},\frac{6}{17},\frac{6}{20},\frac{6}{23}.$
$22$ times the third term will fetch $22\times \frac{6}{11}=12.$