To solve this problem, we need to find the number of terms in the geometric sequence $9, 6, 4, \ldots, \frac{64}{81}$.

Step-by-Step Solution:

1. Identify the first term ($a$) and common ratio ($r$):

   • The first term, $a = 9$.
   • The common ratio $r$ is the ratio of any term to its preceding term. Let's calculate $r$ using the first two terms:

   $r = \frac{6}{9} = \frac{2}{3}$

2. General formula for the $n$-th term of a geometric sequence:

   The formula for the $n$-th term ($a_n$) of a geometric sequence is:

   $a_n = a \cdot r^{n-1}$

3. Set up the equation using the given last term:

   The last term given is $\frac{64}{81}$. We set this equal to the general formula:

   $\frac{64}{81} = 9 \cdot \left(\frac{2}{3}\right)^{n-1}$

4. Solve for $n$:

   • First, divide both sides by 9:

   $\frac{64}{81} \div 9 = \left(\frac{2}{3}\right)^{n-1}$

   • Simplify the left side:

   $\frac{64}{81} \times \frac{1}{9} = \frac{64}{729}$

   So,

   $\left(\frac{2}{3}\right)^{n-1} = \frac{64}{729}$

   • Notice that $64 = 2^6$ and $729 = 3^6$. Therefore, we can express $\frac{64}{729}$ as:

   $\frac{64}{729} = \left(\frac{2}{3}\right)^6$

   • Now set the exponents equal since the bases are the same:

   $n-1 = 6$

   • Solving for $n$, we get:

   $n = 6 + 1 = 7$

Conclusion:

The number of terms in the geometric sequence is 7.

Would you like further details on any step or have any questions?

Here are some related questions you might find helpful:

1. What is a geometric sequence and how is it different from an arithmetic sequence?
2. How do you determine the common ratio in a geometric sequence?
3. What are some real-world applications of geometric sequences?
4. How do you find the sum of a geometric sequence?
5. How can you solve geometric sequences if given different forms of terms (e.g., decimals, fractions)?

Tip: Always double-check your work, especially when simplifying fractions or working with exponents, to ensure accuracy in solving sequence problems.