Find the three terms between asub1=two third and asub5=54

We are tasked with finding three terms between \( a_1 = \frac{2}{3} \) and \( a_5 = 54 \) in a geometric sequence.

### Step 1: Formula for the nth term of a geometric sequence
The general formula for the nth term of a geometric sequence is:

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

Where:
- \( a_n \) is the nth term,
- \( a_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.

### Step 2: Setup the equation for the 5th term
Since we know \( a_5 = 54 \), we can use this to find the common ratio \( r \). The equation for the 5th term is:

\[
a_5 = a_1 \cdot r^{5-1} = a_1 \cdot r^4
\]

Substitute the known values:

\[
54 = \frac{2}{3} \cdot r^4
\]

### Step 3: Solve for the common ratio \( r \)

\[
r^4 = \frac{54}{\frac{2}{3}} = 54 \cdot \frac{3}{2} = 81
\]

Now, take the fourth root of both sides:

\[
r = \sqrt[4]{81} = \sqrt[4]{3^4} = 3
\]

So, the common ratio is \( r = 3 \).

### Step 4: Find the missing terms
Now that we know the common ratio \( r = 3 \), we can find the missing terms. The terms between \( a_1 = \frac{2}{3} \) and \( a_5 = 54 \) are \( a_2 \), \( a_3 \), and \( a_4 \).

1. **\( a_2 = a_1 \cdot r \)**

\[
a_2 = \frac{2}{3} \cdot 3 = 2
\]

2. **\( a_3 = a_2 \cdot r \)**

\[
a_3 = 2 \cdot 3 = 6
\]

3. **\( a_4 = a_3 \cdot r \)**

\[
a_4 = 6 \cdot 3 = 18
\]

### Conclusion:
The three terms between \( a_1 = \frac{2}{3} \) and \( a_5 = 54 \) are:

\[
2, 6, 18
\]

Would you like more details or have any questions?

Here are five related questions to consider:
1. How do you determine the common ratio in an arithmetic sequence?
2. What happens when the common ratio is a fraction?
3. Can you use this method for arithmetic sequences, or does it differ?
4. What if \( a_1 \) is negative—how would that affect the sequence?
5. How does the geometric sequence change if the common ratio is less than 1?

**Tip:** In geometric sequences, if the common ratio is greater than 1, the terms increase exponentially, while if it is between 0 and 1, they decrease.

Learn how to find the three missing terms in a geometric sequence where the first term is 2/3 and the fifth term is 54. This problem covers key concepts such as the geometric sequence formula, exponents, and calculating the common ratio. Suitable for students in Grades 8-10.

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