The given sequence is:

6, 18, 54, 162, ...

To determine the pattern, let's examine the relationship between consecutive terms.

1. **Calculate the ratio between consecutive terms**:
   - \( \frac{18}{6} = 3 \)
   - \( \frac{54}{18} = 3 \)
   - \( \frac{162}{54} = 3 \)

Each term is obtained by multiplying the previous term by 3. This indicates that the sequence is a geometric sequence with a common ratio of 3.

2. **General Formula**:
   The general form of a geometric sequence is given by:

   \[
   a_n = a_1 \times r^{(n-1)}
   \]

   Here:
   - \( a_1 = 6 \) (the first term)
   - \( r = 3 \) (common ratio)

   Therefore, the \(n\)-th term of the sequence can be expressed as:

   \[
   a_n = 6 \times 3^{(n-1)}
   \]

3. **Next Term Calculation**:
   To find the next term after 162:

   - The current term (162) is the 4th term (\( a_4 \)).
   - The next term (\( a_5 \)) would be:

   \[
   a_5 = 162 \times 3 = 486
   \]

So, the next term in the sequence is 486.

### Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What would be the 10th term of this sequence?
2. How do you find the sum of the first \(n\) terms of a geometric sequence?
3. What is the formula for the sum to infinity of a geometric series, and when does it apply?
4. How can you determine if a sequence is geometric or arithmetic?
5. What happens if the common ratio of a geometric sequence is a fraction?

**Tip:** When identifying the type of sequence, always check the difference or ratio between consecutive terms. For arithmetic sequences, the difference is constant, while for geometric sequences, the ratio is constant.

Explore the pattern and formula of the geometric sequence 6, 18, 54, 162, ... Learn how to identify and calculate terms in a geometric sequence with a common ratio of 3. Suitable for Grades 9-12.

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