The sequence given is: 4, -12, 36.

This sequence appears to follow a pattern that can be identified by determining the ratio between consecutive terms. Let's calculate the ratio:

\[ \text{Common ratio} (r) = \frac{\text{second term}}{\text{first term}} = \frac{-12}{4} = -3 \]

\[ \text{Common ratio} (r) = \frac{\text{third term}}{\text{second term}} = \frac{36}{-12} = -3 \]

Since the ratio is consistent, we confirm that this is a geometric sequence with a common ratio of \(-3\).

### General Term of the Sequence
The general term \(a_n\) of a geometric sequence can be found using:
\[ a_n = a_1 \times r^{(n-1)} \]

where:
- \(a_1\) is the first term of the sequence (4),
- \(r\) is the common ratio (\(-3\)),
- \(n\) is the term number.

So, the general formula for the nth term of this sequence is:
\[ a_n = 4 \times (-3)^{(n-1)} \]

### Examples:
- For \(n = 1\):
  \[ a_1 = 4 \times (-3)^{(1-1)} = 4 \times 1 = 4 \]

- For \(n = 2\):
  \[ a_2 = 4 \times (-3)^{(2-1)} = 4 \times (-3) = -12 \]

- For \(n = 3\):
  \[ a_3 = 4 \times (-3)^{(3-1)} = 4 \times (-3)^2 = 4 \times 9 = 36 \]

To find the next term (\(n = 4\)):
- For \(n = 4\):
  \[ a_4 = 4 \times (-3)^{(4-1)} = 4 \times (-3)^3 = 4 \times (-27) = -108 \]

Do you want more details or have any questions?

Here are some related questions:

1. What is the sum of the first 4 terms of this sequence?
2. Can you find the 5th term of the sequence?
3. How do you find the sum of a geometric series?
4. What happens if the first term is changed to a different value?
5. How would you find the sum of an infinite geometric series with a common ratio of less than 1?
6. What is the difference between a geometric sequence and a geometric series?
7. How would the sequence change if the common ratio was positive instead of negative?
8. Can you explain the convergence or divergence of this geometric series?

### Tip
When identifying patterns in sequences, checking the consistency of the ratio or difference between terms helps to determine whether a sequence is geometric or arithmetic.

Explore the pattern and calculation of terms in the geometric sequence 4, -12, 36. Learn how to identify the common ratio, calculate the nth term, and find subsequent terms using the general formula. Suitable for high school students studying geometric sequences.

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