1/3 and 64/3 find the 1 geometric mean with the solution an=a1 rn-1 

To find the geometric mean between two numbers, you use the formula:

\[
\text{Geometric Mean} = \sqrt{a_1 \times a_2}
\]

However, since the problem asks for a solution using the formula for the \(n\)th term of a geometric sequence, \(a_n = a_1 \cdot r^{n-1}\), we'll proceed as follows:

### Step 1: Identify the first term (\(a_1\)) and the last term (\(a_n\))
- \(a_1 = \frac{1}{3}\)
- \(a_2 = \frac{64}{3}\) (since we are finding the 1st geometric mean between these two values)

### Step 2: Find the common ratio (\(r\))
The formula for the \(n\)th term in a geometric sequence is:

\[
a_n = a_1 \cdot r^{n-1}
\]

For \(n = 2\) (since the geometric mean is the second term):

\[
a_2 = a_1 \cdot r^{2-1} = a_1 \cdot r
\]

So, we can solve for \(r\):

\[
r = \frac{a_2}{a_1} = \frac{\frac{64}{3}}{\frac{1}{3}} = 64
\]

### Step 3: Calculate the geometric mean (\(a_2\))
The geometric mean is the second term in the sequence:

\[
a_2 = a_1 \cdot r = \frac{1}{3} \cdot 64 = \frac{64}{3}
\]

Thus, the geometric mean between \(\frac{1}{3}\) and \(\frac{64}{3}\) is:

\[
\boxed{\frac{64}{3}}
\]

This confirms that \(r = 64\) was correct and that the geometric mean is indeed \(\frac{64}{3}\).

Would you like more details or have any questions?

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**Related Questions:**

1. How do you find the geometric mean between three numbers?
2. What is the difference between arithmetic and geometric sequences?
3. How do you determine the number of terms in a geometric sequence?
4. How is the sum of a geometric series calculated?
5. Can you find the common ratio if only the first and last terms are given?

**Tip:** When dealing with sequences, ensure you correctly identify whether it's arithmetic or geometric to apply the right formulas.

1/3 and 64/3 find the 1 geometric mean with the solution an=a1 rn-1

Learn how to find the geometric mean between 1/3 and 64/3 using geometric sequence formulas. This problem explores algebraic concepts like common ratio and properties of geometric sequences, suitable for students in Grades 8-10.

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