Question

Insert 5 geometric means between 16 and $\frac{1}{4}$.

Answer 1

5 Geometric means between 16 and $\frac{1}{4}$

Let $G_1,G_2,G_3,G_4,G_5,G_6$ be 5 geometric means between 16 and $\frac{1}{4}$

$\therefore 16,G_1,G_2,G_3,G_4,G_5,\frac{1}{4}$ is a G.P. with

$a=16,b=\frac{1}{4}$

Then,

$r={\left(\frac{b}{a}\right)}^{\frac{1}{n+1}}$

$={\left(\frac{\frac{1}{4}}{16}\right)}^{\frac{1}{5+1}}={\left(\frac{1}{2^6}\right)}^{\frac{1}{6}}=\frac{1}{2}$

$\therefore G_1=ar=16\times \frac{1}{2}=8$

$G_2=a{r}^2=16\times \frac{1}{4}=4$

$G_3=a{r}^3=16\times \frac{1}{8}=2$

$G_4=a{r}^4=16\times \frac{1}{16}=1$

$G_5=a{r}^2=16\times \frac{1}{2^5}=\frac{1}{2}$

Hence, 8, 4, 2, 1, $\frac{1}{2}$ are 5 geometric means

between 16 and $\frac{1}{4}$.