Insert 5 geometric means between 16 and
Question:

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

Solution:

Let the $5 \mathrm{G} . \mathrm{M} . \mathrm{s}$ betweem 16 and $\frac{1}{4}$ be $\mathrm{G}_{1}, \mathrm{G}_{2}, \mathrm{G}_{3}, \mathrm{G}_{4}$ and $\mathrm{G}_{5}$.

$16, \mathrm{G}_{1}, \mathrm{G}_{2}, \mathrm{G}_{3}, \mathrm{G}_{4}, \mathrm{G}_{5}, \frac{1}{4}$

$\Rightarrow a=16, n=7$ and $a_{7}=\frac{1}{4}$

$\because a_{7}=\frac{1}{4}$

$\Rightarrow a r^{6}=\frac{1}{4}$

$\Rightarrow r^{6}=\frac{1}{4 \times 16}$

$\Rightarrow r^{6}=\left(\frac{1}{2}\right)^{6}$

$\Rightarrow r=\frac{1}{2}$

$\therefore G_{1}=a_{2}=a r=16\left(\frac{1}{2}\right)=8$

$\mathrm{G}_{2}=a_{3}=a r^{2}=16\left(\frac{1}{2}\right)^{2}=4$

$\mathrm{G}_{3}=a_{4}=a r^{3}=16\left(\frac{1}{2}\right)^{3}=2$

$\mathrm{G}_{4}=a_{5}=a r^{4}=16\left(\frac{1}{2}\right)^{4}=1$

$\mathrm{G}_{5}=a_{6}=a r^{5}=16\left(\frac{1}{2}\right)^{5}=\frac{1}{2}$