# How do you write a sequence that has three geometric means between 256 and 81?

Nov 20, 2015

$256 , 192 , 144 , 108 , 81$

#### Explanation:

$256 = {4}^{4}$ and $81 = {3}^{4}$

So the geometric mean of $256$ and $81$ is:

$\sqrt{256 \cdot 81} = \sqrt{{4}^{4} \cdot {3}^{4}} = {4}^{2} \cdot {3}^{2} = 144$

The geometric mean of $256$ and $144$ is:

$\sqrt{256 \cdot 144} = \sqrt{{4}^{4} {4}^{2} {3}^{2}} = {4}^{3} \cdot 3 = 192$

The geometric mean of $144$ and $81$ is:

$\sqrt{144 \cdot 81} = \sqrt{{4}^{2} {3}^{2} {3}^{4}} = 4 \cdot {3}^{3} = 108$

The sequence: $256 , 192 , 144 , 108 , 81$ is a geometric sequence with common ratio $\frac{3}{4}$