Question

Insert $3$ geometric means between $2\frac{1}{4}$ and $\frac{4}{9}$.

Answer 1

let $a=2\frac{1}{4}=\frac{9}{4},b=\frac{4}{9}$
$3$ G.M between $a$ and $b$ are given by
$ar,a{r}^2,a{r}^3$
$b=a{r}^4=\frac{4}{9}$
$b=r^4=\frac{16}{81}$
$r=\frac{2}{3}$
Thus, the $3$ G.M between $a$ and $b$ are
$\frac{9}{4}\times \frac{2}{3},\frac{9}{4}\times \frac{4}{9},\frac{9}{4}\times \frac{8}{27}$
$\frac{3}{2},1,\frac{2}{3}$