Question

Insert $5$ geometric means between $3\frac{5}{9}$ and $40\frac{1}{2}$.

Answer 1

Let $a=3\frac{5}{9}=\frac{32}{9},b=40\frac{1}{2}=\frac{81}{2}$
$5$ G.M between $a$ and $b$ are given by
$ar,a{r}^2,a{r}^3,a{r}^4,a{r}^5$
$b=a{r}^6=\frac{81}{2}$
$b=r^6=\frac{3^6}{2^6}$
$r=\frac{3}{2}$
Then, $5$ G.M between $a$ and $b$ are
$\frac{32}{9}\times \frac{3}{2},\frac{32}{9}\times \frac{9}{4},\frac{32}{9}\times \frac{27}{8},\frac{32}{9}\times \frac{81}{16},\frac{32}{9}\times \frac{243}{32}$
i.e. $\frac{16}{3},8,12,18,27$