To find the 8th term of a geometric sequence, we use the formula for the $n$-th term of a geometric sequence:

$a_n = a_1 \cdot r^{(n-1)}$

where:

• $a_n$ is the $n$-th term.
• $a_1$ is the first term.
• $r$ is the common ratio.
• $n$ is the term number.

For the given sequence $6, 18, 54, 162, \ldots$:

1. The first term $a_1 = 6$.
2. The common ratio $r$ can be found by dividing the second term by the first term:

$r = \frac{18}{6} = 3$

Now, we need to find the 8th term ($a_8$):

$a_8 = 6 \cdot 3^{(8-1)}$

Calculating $3^{7}$:

$3^7 = 2187$

Now, compute $a_8$:

$a_8 = 6 \cdot 2187 = 13122$

Answer:

The 8th term of the geometric sequence is 13,122.

Would you like a detailed explanation of how I calculated this or have any other questions?

Here are 5 related questions to explore further:

1. How do you find the sum of the first $n$ terms of a geometric sequence?
2. What is the 10th term of a geometric sequence where the first term is 5 and the common ratio is 2?
3. How do you determine if a sequence is geometric?
4. Can you derive the formula for the $n$-th term of a geometric sequence?
5. How would you find the common ratio if only the 3rd and 5th terms of a geometric sequence are given?

Tip: In geometric sequences, the common ratio remains constant throughout the sequence.