what is the 8th term of the geometric sequence 6,18,54,162

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.

Learn how to calculate the 8th term of the geometric sequence 6, 18, 54, 162 using the formula a_n = a_1 * r^(n-1). This solution explains the step-by-step process of finding the common ratio and applying the formula, ideal for students in Grades 9-12.

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