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.