To insert three numbers between 6 and $\frac{32}{27}$ such that the five numbers form a geometric sequence, we need to follow the process of finding the common ratio of the sequence and the intermediate terms.

Steps:

1. Set up the sequence:

   Let the geometric sequence be:

   $6, \, a_1, \, a_2, \, a_3, \, \frac{32}{27}$

   Here, $a_1$, $a_2$, and $a_3$ are the three numbers we need to find. The first term is 6, and the last term is $\frac{32}{27}$.

2. Use the formula for a geometric sequence:

   In a geometric sequence, the ratio between consecutive terms is constant. Let the common ratio be $r$. Then, the sequence follows this pattern:

   $a_1 = 6r, \quad a_2 = 6r^2, \quad a_3 = 6r^3, \quad \frac{32}{27} = 6r^4$

3. Find the common ratio $r$:

   The last term of the sequence gives the equation:

   $6r^4 = \frac{32}{27}$

   Solve for $r$:

   $r^4 = \frac{32}{27} \div 6 = \frac{32}{27} \times \frac{1}{6} = \frac{32}{162} = \frac{16}{81}$

   Taking the fourth root of both sides:

   [ r = \sqrt[4]{\frac{16}{81}} = \frac{\sqrt[4]{16}}{\sqrt[4]{81}} = \frac{2}{3} ]

4. Find the three numbers:

   Now that we have the common ratio $r = \frac{2}{3}$, we can find the three geometric means:

   $a_1 = 6r = 6 \times \frac{2}{3} = 4$ $a_2 = 6r^2 = 6 \times \left( \frac{2}{3} \right)^2 = 6 \times \frac{4}{9} = \frac{24}{9} = \frac{8}{3}$ $a_3 = 6r^3 = 6 \times \left( \frac{2}{3} \right)^3 = 6 \times \frac{8}{27} = \frac{48}{27} = \frac{16}{9}$

5. Final sequence:

   The full geometric sequence is:

   $6, \, 4, \, \frac{8}{3}, \, \frac{16}{9}, \, \frac{32}{27}$

Thus, the three geometric means between 6 and $\frac{32}{27}$ are $4$, $\frac{8}{3}$, and $\frac{16}{9}$.

---

Would you like more details on any specific step or clarification?

Relative Questions:

1. How do you derive the formula for the common ratio in a geometric sequence?
2. What is the difference between arithmetic and geometric sequences?
3. How can you generalize the insertion of geometric means for any two numbers?
4. How do you calculate roots of fractions such as the fourth root of $\frac{16}{81}$?
5. What are some real-world applications of geometric sequences?

Tip:

In a geometric sequence, each term is a constant multiple of the previous term, unlike in arithmetic sequences where the difference between terms is constant.